Minfeub had an emergency screening of Darren Aronofsky’s debut Pi this week since sable was on break and needed a film. rachel suggested this one because sable couldn’t stop yapping about Fractal Block World, a mathematical artwork from this decade
But when we start from the Repulsion of the determinately existing Ones, and in doing so posit Attraction as being added to them from without, then, though inseparable, both are kept apart as distinct determinations; we have seen, however, that Attraction does not only presuppose Repulsion, but that there also is a reverse relation of Repulsion to Attraction, so that the former equally has its presupposition in the latter.
According to this determination they are inseparable, and also each is determined as Ought and Barrier relatively to the other. Their Ought is their abstract determinateness as being in themselves; thereby, however, it is just referred beyond itself and relates itself to the other, so that each is mediated
by the other as other. Their independence consists in this, that in this mediation they are posited as another determining for each other (Repulsion is the positing of the Many, Attraction of the One ; the latter is also negation of the Many, the former, the negation of their ideality in the One), and in that Attraction is Attraction only by the mediation of Repulsion, and Repul¬ sion Repulsion only by the mediation of Attraction. The further fact that mediation through Other is here in fact rather negated and each of these determinations is self-mediation, results from closer contemplation and carries them back to the unity of their concept.
First, then, the attitude of Repulsion and Attraction (which as yet are only relative) already implies that each presupposes itself, and in so doing relates itself only to itself.
Relative Repulsion is the mutual repelling of the many given Ones, which are supposed to exist immediately. But that there are many Ones, this assertion is just Repulsion; and any pre¬ supposition of Repulsion is only its own positing. Further, the determination of Being, which might belong to the Ones apart from the fact that they are posited — a determination which would give them pre-r?dstence — also belongs to Repulsion. It is by means of Repulsion that the Ones manifest and preserve themselves as Ones, and have Being as such. Their Being is Repulsion itself, which thus is not relative to any other Deter¬ minate Being, but stands related only to itself.
Attraction is the positing of the One as such, of the real One, in opposition to which the Many are determined, in their Determinate Being, as only of ideal nature, and as vanishing. And thus Attraction also presupposes itself, namely in the determination of the other Ones to be of ideal nature, which elsewhere are supposed to be repellent for-themselves and for- others, and therefore also for whatever attracts. They gain ideality as opposed to this determination of repulsion, not in the end and only through their relation to Attraction; rather it is presupposed, and is the self-existent ideality of the Ones, since, as Ones (and here we include the One which is imagined as attracting), they are undistinguished one from another, they are one and the same.
If then the two determinations thus presuppose themselves, sack fof itself this further means that each contains the other
as moment. Self-presupposition is the positing of self in One as its own negative, — here we have Repulsion; and that which is presupposed is the same as that which presupposes, — here we have Attraction. The fact that each is in itself mere moment constitutes the transition of one into the other, negating itself in itself and positing itself as its own Other. The One as such is self-transcendence, and its only meaning is to posit itself as its Other, the Many ; and the Many similarly is only collapse upon itself, — it posits itself as its Other, the One, and in it alone relates itself to itself; so that each continues itself in its Other; and thus we have already here self-transcendence (Repulsion) and self-positing-as-One (Attraction) present un¬ separated. That which is posited in a Repulsion and Attraction which are relative (that is, which presuppose immediate Ones existing determinately) is this, that each is in itself its own negation, and therefore the continuation of its self into its Other. The Repulsion of Ones having Determinate Being is the self-preservation of each one by means of the mutual repulsion of the others: thus (1) the other Ones are negated in it ; this is the aspect of its Determinate Being, or Being-for- Other; this therefore is Attraction as being the ideality of the Ones : — and (2) the One is in itself, without relation to others ; but “in-itself” passed long ago into “Being-for-self,” and further the One in itself is by its determination that Becoming of Many. — The Attraction of determinately existing Ones is their ideality, and is the positing of the One, wherein as negation and production of the One it thus transcends itself, and, as positing of the One, is its own negative, or Repulsion.
Thus Being-fbr-Self has completed its development and has reached its result. The One as in infinite self-relation (infinite because it is the posited negation of negation) is mediation: itself, as its own absolute (that is, abstract) otherness (the Many), it repulses from itself; and, entering into a negative relation to this its not-Being, it transcends it and thus again is self¬ relation; and One is no more than this Becoming, in which the determination has vanished which states it to be Beginning, that is, posits it as immediate and having Being, and, further, as reconstituting itself, in the result, as One, a One equally immediate and exclusive. The process, which is the One, everywhere posits and contains it only as transcended This
transcendence first is determined only as relative, as a relation to other Determinate Being, which thus itself is a different Repulsion and Attraction; next, it shows itself as pacing over into the infinite relation of mediation through negation of the external relations of Beings immediate and Determinate, having for result this Becoming which, through the instability of its moments, is the collapse, or rather self-gathering, into simple immediacy. Having acquired this new determination, this Being is Quantity.
i think pi is about giving up, and fractal block world is about the tragedy of loving the person who never gave up, totally lost it, and might not be able to regain it. in fractal block world you’re the lady friend [sic; the lady friend was a character in Pi who gave max, the main math man, samosas etc…]
If we review in brief the moments of this transition of Quality into Quantity, the qualitative has for fundamental determination Being and immediacy, where limit and deter¬ minateness are identical with the being of the Something in such a manner that, these being altered, the Something itself vanishes. Being posited in this manner it is determined as finite. By virtue of the immediacy of this unity, in which distinction has disappeared, though it is in itself present in the unity of Being and Nothing, the distinction becomes Otherness in general and is outside this unity. This relation to Other is in contradiction to the immediacy in which qualitative deter¬ minateness is self-relation. This Otherness transcends itself in the infinity of Being-for-Self; by the latter the distinction, which it has in and by itself in the negation of negation, is made real into the stages of One and Many and their relations : the qualitative is elevated to the unity which is true because it is no longer immediate but posited as in harmony with itself.
This Unity then is (a) Being, purely affirmative, that is, immediacy mediated with itself through the negation of nega¬ tion; Being is posited as the unity which runs through each determinateness, through limit and so on; these are posited in it as transcended :
(fi) Determinate Being; according to such a determination it is negation or determinateness as moment of affirmative Being; but it is such no longer as immediate, but as intro- reflected, related not to Other but to Self; it is Determined- ness-in-itself, or One ; Otherness as such is itself Being-for-Self:
(y) Being-for-Self, as this Being which endures through determinateness; here One and Determinedness-in-itself are themselves posited as transcended. The One is now determined as haying passed beyond itself and as Unity; and hence the
One, the absolutely determinate Limit, is posited as the Limit which is not Limit, and as indifferent to Being although it is posited in it.
Attraction and Repulsion are generally, of course, regarded as forces. This determination, and the relations connected with it, should be compared with the concepts which they have been found to be. — In the view mentioned they are regarded as independent, as if they were not related to one another by their own nature; that is, they are taken not as two moments passing each into its opposite, but rather as standing one against the other in fixed opposition. It is further imagined that they meet in a third element, namely, matter; but this takes place in such a manner that this unification is not held to be their truth: rather, each is a first principle and is for itself, while matter arid its determinations are posited and created by them. When it is said that matter contains the forces, this unity is intended to mean a connexion, while at the same time they are presupposed as having independent self-existence.
Kant, as we know, has constructed matter from Repulsive and Attractive force, or, at least, he has erected (as he calls it) the metaphysical elements of this construction. — It will not be without some interest to cast light upon this construction. This metaphysical exposition of an object which, not only itself but also in its determinations, appeared to belong only to experience, is remarkable : first, because, as an essay of the concept, it furnished at least the impulse to the modem philosophy of nature, which does not make nature as a sense- datum the foundation of science, but proceeds from the absolute concept to the cognition of its determinations ; next, because often a halt is made at this Kantian construction, and it is taken as the philosophic beginning and foundation of physics.
Now an existent such as empirical matter is not the object of Logic any more than are Space and the determinations of Space. But Repulsive and Attractive force, too, in so far as they are looked on as forces of empirical matter, are based upon the pure determinations of One and Many here considered, and on those relations of them which (because these names were most obvious) I have called Repulsion and Attraction.
Closely considered, Kant’s method in the deduction of matter from these forces, which he calls a construction, does not deserve this name, unless indeed every kind of reflection, even analytical reflection, be called a construction; and, indeed, later natural philosophers have given the name of construction to the shallowest reasoning and the most baseless concoction of arbitrary imagination and thoughtless reflection — especially if it employed and everywhere displayed the so-called factors of attractive and repulsive force.
For at bottom Kant’s method is analytical and not con¬ structive. He presupposes the idea of matter, and next asks what forces are necessary to maintain its determinations thus assumed. Thus first he postulates Attractive force, because really matter cannot exist through Repulsion alone without Attraction ( Anfangsgriinde der Naturwissenschaft, pp. 53 et seq.). Next he derives Repulsion, too, from matter, and gives as reason the fact that we imagine matter as impenetrable, since it presents itself under this determination to the sense of feeling, by which it manifests itself to us. Consequently (he proceeds) Repulsion is immediately thought in the concept of matter, since it is immediately given therein; whereas Attraction is syllogistically added to it. But these syllogisms, too, are based on what has just been said, namely, that matter having only repulsive force does not exhaust our image of matter. — It is evident that this is the proceaure of knowledge reflecting on experience, which first perceives determinations in the pheno¬ menon, next makes them the basis, and finally assumes for their so-called explanation corresponding fundamental materials or forces which are supposed to produce these determinations of the phenomenon.
-sable
In view of the different fashion in which repulsive and attractive force are thus discovered by cognition in matter, Kant further observes that attractive force really equally belongs to the concept of matter, “although it is not contained in it.” Kant lays emphasis on this last expression. Yet it is not clear what the distinction may be ; for a determination which belongs to the concept of a thing must be truly contained in it.
What causes the difficulty and gives rise to this vain subter¬ fuge is that Kant from the beginning arbitrarily counts im- penetr ability alone as proper to the concept of matter; this,
he says, we perceive by the senses, and therefore repulsive force — the repelling of other from one — is immediately given. But if he goes on to say that matter cannot exist without attraction, then this assertion is based on a view of matter which is taken from experience; perception therefore must yield us the deter¬ mination of attraction, too. Indeed it is easy to perceive that beyond its Being-fbr-Self, which transcends Being-for-Other (or offers resistance), matter contains a relation of entities which are for themselves, and thus has spatial extension and cohesion, which latter, in the qualities of rigidity and solidity, is very strong. Explanatory physics demands, in order (for instance) that a body may be tom apart, a force which must be stronger than the mutual attraction of the parts of that body. And from this truth reflection can just as immediately deduce attractive force, or take it as given, as it did with repulsive force. Indeed, if we consider Kant’s syllogisms from which he tries to deduce attractive force (the proof of the proposition, that the possibility of matter demands attractive force as a second basic power, loc. cit.), they are found to con¬ tain nothing save that mere repulsion would not make matter spatial. When matter is presupposed as filling space, continuity is ascribed to it, and the basis of continuity is supposed to be attractive force.
But though this so-called construction of matter would at most have value as analysis (a value diminished by the impure exposition), still the fundamental thought which derives cog¬ nition of matter from these two opposed determinations as from its basic forces, must always be highly esteemed. Kant chiefly is at pains to banish the vulgar and mechanistic idea which stops at the one determination, namely, impenetrability, or self-existent puncticity, and makes into something external the opposite determination, the relation of matter in itself or in other matters, which in turn are regarded as separate Ones ; a manner of presentation which, as Kant says, will allow no other motive forces except pressure and impact, that is, action from without. This externality of cognition always presupposes motion as already existing externally in matter, and never thinks of taking it as something internal and of comprehending it in matter, which thus is assumed in itself as motionless and inert. This point of view looks only on common meshanics,
and not on immanent and free movement. — It is true that Kant transcends this externality in so far as Attraction (or the relation of one matter to another in so far as they are taken as separate, of else of matter in general, in its existence beyond itself) is with him made into a force of matter itself; still, on the other side, his two basic forces remain, within matter, opposed to one another, external and independent.
The independent distinction between these two factors, attri¬ buted to them from the point of view of this cognition, was void, and equally void every other distinction must show itself to be if it is made with regard to the determination of then- content as something which ought to be fixed ; for, as they were considered above in their truth, they are no more than moments, which pass into one another. — I will consider these further determinations of distinction as indicated by Kant.
Kant determines attractive force as a penetrative force, by means of which one piece of matter can act immediately upon parts of another even beyond the surface of contact; repulsive force, on the other hand, he determines as a superficial force, by means of which parts of matter can act upon one another only at the common surface of contact. The reason cited for this determination of the latter is as follows : — “Parts in contact with one another limit each the sphere of activity of the other, and repulsive force cannot move any more distant part except through those parts which intervene; an immediate action of one part of matter on another, passing straight through these, by means of forces of expansion (that is, repulsive forces), is impossible.” (See ibid. Erklarung und Zuscitze, p. 67.)
And here the reader must remember this: when “nearer” and “more distant” parts of matter are assumed, the distinction would also arise with regard to Attraction, in this way, that though one atom might act upon another, yet a third and more distant atom (between which and the attracting atom the “other” atom would be) would next enter into the sphere of attraction of the intermediate (and more proximate) atom; thus the action of the first atom on the third would not be immediate and simple. Thus mediated action would result for attractive as well as for repulsive force. Further, true penetra¬ tion of attractive force would then have to consist solely in the fact that all parts of matter had power to attract in themselves.
and not in the passivity of a certain number and the activity of just one atom. — With immediate regard to repulsive force itself we may remark that in the passage quoted there occur “parts in contact with one another”; which means a homo¬ geneity and continuity of finished matter not admitting of repulsion. But this homogeneity of matter, where parts, no longer separated by void, are in contact with one another, already presupposes the transcendence of repulsive force; according to the sense-image of Repulsion which governs here, parts in contact must be taken as such as do not repel one another. The tautological conclusion therefore is that where the not-being of Repulsion is assumed there can take place no Repulsion. But nothing further follows from this by way of a determination of repulsive force. — If, however, it is considered that parts in contact touch one another only in so far as they still keep one another apart, then repulsive force no longer resides merely in the surface of matter, but within that sphere which was intended to be the sphere of Attraction only.
Next Kant assumes the determination that “through attrac¬ tive force matter occupies but does not fill space” (ibid.) ; “and since matter does not through this force fill space, it can act across empty space, since there is no intervening matter which could limit it.” — This distinction resembles the one mentioned above, where a determination is supposed to belong to but not to be contained in the concept of a thing; similarly here matter is to occupy space but not to fill it. It is then Repulsion (if we remain at its first determination) by means of which the Ones repulse one another and are related to one another only negatively, that is, here, across empty space. But here it is attractive force which keeps space empty ; it does not fill space by means of the relation between atoms which here prevails ; that is, it keeps the atoms in a negative relation to one another. — Thus Kant is unconsciously brought to do what is implicit in the nature of the case, that is, he ascribes to attractive force precisely what he ascribed (according to the first deter¬ mination) to the opposite force. While the distinction between the two forces was being laid down, one had passed over into the other. — By Repulsion, on the other hand, matter is intended to fill space, that is, through it the empty space left over by Attractive force is intended to vanish. And in fact, in cancelling
math man was hanging out and then had a problem with his MATH MACHINE; there was an ant and it was eating the MATH MACHINE so he bought a new computer and ate the MATH GOO to become MATHIER and then the rabbi and the math demons started fighting him for the math but math man did a LOBOTAMY to kill the math and now we will never know the math number which is definitely not 9414324343151265932105487239048682851291347487602767195923460238582958304725016523252592969257276553643634627271840120126431475463294501278472648410756223478962672859285829534750277226264645621761398482951947512398501
empty space it cancels the negative relation of the atoms or Ones, that is, the Repulsion which takes place between them; in other words, Repulsion is determined as its own opposite.
Thus the distinctions are blurred; and to this is added the confusion arising from the fact that (as was observed at the beginning) Kant’s exposition of the opposed forces is analytic, and in the whole course thereof matter, which should be derived first from its elements, appears fully and completely constituted. In the definition of superficial and of penetrative force both are taken as motive forces, by means of which matter can act in one or in the other manner. — They are thus here set out as forces by which matter is not created but, being already given, is only moved. But in so far as we speak of forces by means of which matter acts upon matter, or moves itself, this is a very different thing from the determination and relation which they were supposed to have as moments of matter.
In a further determination centripetal and centrifugal force form the same antithesis as attractive and repulsive force. The former seem to offer an essential distinction, since in their sphere one One, or centre, is fixed, relatively to which other Ones behave as not self-existent, so that the distinction between the forces may be connected to this presupposed distinction between a central One and others which are not independent relatively to it. But in so far as they are used for explanation, for which purpose (like Repulsive and Attractive force) they are taken to be in inverse quantitative ratio, one increasing as the other decreases, the phenomenon of movement and its inequality, which they are accepted in order to explain, is supposed only to result from them. However, it is only necessary to examine any exposition of a phenomenon, like the unequal velocity of a planet in its orbit around the central body, to see that, when based on the opposition of these forces, it is full of confiision and that it is impossible to disentangle the magni¬ tudes ; so that the one, which in the exposition is assumed to be diminishing, may just as well be assumed to increase, and con¬ versely. To make this evident would require an exposition more lengthy than can be offered here; what is essential occurs below, under the category of Inverse Ratio.
We have indicated the difference between Quantity and Quality. Quality is primary and immediate determinateness; quantity is such determinateness as has become indifferent to Being. It is a limit which also is no limit, it is Being-for-Self which is simply identical with Being-for-Other, the Repulsion of many Ones which immediately is their Non-Repulsion or continuousness.
Now Being-for-Self has been so posited as not to exclude its Other but rather affirmatively to continue itself into it ; hence it is Otherness (in so far as Determinate Being again arises in this continuity) and its determinateness simultaneously ceases to be simply self-related and is no longer the immediate deter¬ minateness of the determinately existing Something; instead, it is posited as self-repelling and as having its relation to itself, as determinateness, rather in some other Determinate Being (which is for itself) ; and, since also they exist as indifferent, intro-reflected, and relationless limits, determinateness is simply beyond itself; it is external to itself and is a Something as equally external; such a limit, together with the indifference of itself to itself and to Something, constitutes its quantitative determinateness.
Pure Quantity must here be distinguished from determinate quantity, or Quantum. As the former, Quantity is, first, real Being-for-Self, which has returned upon itself and as yet has no determinateness ; as infinite, homogeneous unity which con¬ tinues into itself.
Secondly, this passes over into a determinateness, which is posited of it as one which is none, or, rather, is only external. It becomes Quantum. Quantum is indifferent determinateness, that is, it passes beyond and negates itself; as this otherness of otherness it is caught in the infinite progress. Infinite Quan¬ tum, however, is the transcendence of indifferent determinate¬ ness, it is the rehabilitation of Qpality.
Thirdly, Quantum in qualitative form is quantitative relation. Quantum merely passes beyond itself; in Relation it passes over into its otherness in such a way that the latter, which forms its determination, is posited simultaneously, and is another Quan¬ tum ; we then find that it has returned to itself and that it is related to itself (namely in its otherness).
Further, this relation is based on the externality of Quantum ; the quanta which are related (that is, whose self-relation con¬ sists in their being beyond themselves) are indifferent. Thus this relation is a merely formal unity of Quality and Quantity. The dialectic of this relation is its transition into their absolute unity, into Measure.
In the Something, its limit, as Quality, essentially is its determinateness. But if by limit we mean quantitative limit, and a field (for instance) changes this its limit, it remains what it was — field. Whereas if its qualitative limit is changed, then its determinateness, which makes it a field, is changed, and it becomes meadow, forest, and so on. — A red which becomes darker or lighter still is red ; but if it were to change its quality, it would cease to be red and would become blue or some other colour. — This determination of magnitude as Quantum as deduced above — namely, an unchanging substratum of Being which is indifferent to its determinateness — can also be deduced from every other example.
good film; the soundtrack was cute and the body horror was satiating; 10/12 (twelve is the best number btw uwu)
By magnitude Quantum is meant, as in the examples cited, and not Quantity ; this is the chief reason why this alien term must be borrowed.
The definition of magnitude which mathematicians employ also refers to Quantum. Magnitude is generally defined as something which may be increased or diminished. But to increase means to make something more great ( maitis ), and to diminish to make it less great (minus). Thus there is a distinction between magnitude in general and itself, and magnitude appears as something of which the magnitude may be varied. The definition thus proves improper, since it employs the determination which it would define. In so far as the same determination must not be employed in it, “more” and “less” must be analysed into an affirmative addition (which, according
to the nature of Quantum, is likewise external) and a sub¬ traction (an equally external negation). Indeed, it is to tbis external kind both of Reality and of Negation that the nature of change determines itself in Quantum. The governing moment, therefore (which is the main matter) must not be mis¬ taken in this incomplete expression; namely, the indifference to change, so that its own More and Less, its indifference to itself, lies in its very concept.
Quantity is Being-for-Self transcended; the repelling One, which (as we saw) remains merely negative towards the ex¬ cluded One, has passed over into relation to it, and, becoming identical with Other, has lost its determination ; Being-for-Self has passed over into Attraction. The absolute brittleness of the repelling One has melted down into this unity ; and this, since it contains this One and also is determined by the immanent Repulsion, is unity with itself as being unity with Being-beyond- self. Attraction thus is the moment of continuity in Quantity.
Continuity thus is simple and self-identical self-relation, interrupted by no limit or exclusion ; not, however, an imme¬ diate unity, but a unity of the Ones which are for themselves. The externality of plurality is here still contained, but as something undifferentiated and uninterrupted. In continuity, plurality is posited as it is in itself ; each of the many is what the others are, each is equal to the other, and hence plurality is simple and undifferentiated equality. Continuity is this moment of the self-identity of Being-external-to-one-another, the self-prolongation of different Ones into other and distinct Ones.
Thus Magnitude in continuity has the moment of discrete¬ ness immediately — Repulsion, which now in Quantity is bare moment. — Stability is self-identity, self-identity of the many which do not become exclusive; it is only Repulsion which expands self-identity into continuity. And thus on its side dis¬ creteness is coalescent discreteness, where the Ones are related not to the void (or negative) but to their own stability, and do not, in the Many, interrupt this self-identity.
Quantity is the unity of these moments, of continuity and discreteness ; but at the outset it is so in the form of one of them, G*
namely of continuity ; and this is the result of the dialectic of Being-for-Self, which has collapsed into the form of self-identical immediacy. Quantity as such is just this simple result, in so far as its moments are not yet developed, nor posited in it. — At present, posited as Being-for-self as it truly is, it only contains them. It was determined as self-transcendent self-relation, a perennial egress beyond itself. But that which it repels is itself; and thus Repulsion is its own creative onward flow. Because of this identity with that which is repulsed, this discreteness is an uninterrupted continuity; and because it passes beyond itself, this continuity, without being interrupted, is at the same time plurality, which equally immediately remains in its identity with itself.
Pure Quantity as yet has no limit, or in other words is not yet Quantum ; and even in so far as it becomes Quantum, limit does not bar it in ; it rather consists in the fact that limit does not bar it in, and that it contains Being-for-Self as tran¬ scended. Discreteness is moment in it, which may be expressed by saying that Quantity always and everywhere is the real potentiality of One, but that also conversely One is equally absolutely continuous.
Imagination operating without concept easily changes Con¬ tinuity into Combination, that is, into an external relation of the Ones to one another, the One remaining in its absolute and brittle exclusiveness. But we saw when examining the One that, in its own true nature, it passes over into its ideality, which is Attraction, and that thus continuity is not external but peculiar to it, and founded in its essence. It is this exter¬ nality of continuity for the One in which atomism remains entangled; it is this which imagination finds it so hard to leave. — On the other hand, mathematics rejects a metaphysic which should be content to allow time to consist of points of time, space in general (or, as a first step, the line) of points in space, the plane, of lines, and the whole of space, of planes; it allows no validity to such discontinuous Ones. And although it determines, for instance, the magnitude of a plane as con¬ sisting of the sum of an infinity of lines, yet this discreteness is taken only as a momentary image; and the infinite plurality
-recima from minfeub (the m in minfeub stands for math [sic; recima from minfeub (the m in minfeub stands for math) left the paren unclosed. rachel notes that closing parens at the end of a message are unnecessary, and when she had to find a bijection from mountain ranges to slowly increasing sequences, (both counted by catalan numbers BTW) this fact came in handy
of lines implies, since the space which they are meant to con¬ stitute is after all limited, that their discreteness has already been transcended.
It is the concept of pure Quantity as opposed to the mere image of it that Spinoza (to whom it was of especial importance) has in mind when he speaks of Quantity as follows (Eth. P. I. Prop. XV, Schol.) : —
Quantitas duobus modis a nobis concipitur, abstracte scilicet sive super* ficialiter, prout nempe ipsam imaginamur; vel ut substantia, quod a solo intellectu fit. Si itaque ad quantitatem attendimus, prout in imaginations est, quod saepe et facilius a nobis fit, reperietur finita, divisibilis, et ex partibus conflata, si autem ad ipsam, prout in intellectu est, attendimus, et cam, quatenus substantia est, concipimus, quod difiicillime fit, — infinita, unica, et indivisibilis reperietur. Quod omnibus, qui inter imaginadonem et intellectum disdnguere sciverint, satis manifestum erit.
Should more definite examples of pure Quantity be demanded. Space and Time, Matter in general, Light, and so forth, the Ego itself, will furnish them; only by Quantity we must not understand (as was remarked above) Quantum. Space, Time, and so on, are expansions or pluralities where egress beyond self, and a constant flow, take place; though what is reached is not the opposite (Quality, or the One), but, passing beyond themselves, they are engaged upon a perpetual self¬ production of their unity.
Space is such an absolute being-beyond-self; it is also absolutely uninterrupted, a still-repeated otherness which is identical with itself; and Time is an absolute passing-beyond- self, a creation of the One (the point of time or Now), which immediately becomes the annihilation of it, and also the per¬ petual annihilation of this destruction : so that this self-creation of Not-being is also simple self-equality and self-identity. — As regards Matter as Quantity, among the seven Prepositions surviving of the first dissertation of Leibniz there is one (the second) which deals with this (left-hand page of Part One of his works) ; it runs thus : Non omnino improbabile est , materiam et quantitatem esse realiter idem. — And, indeed, there is no differ¬ ence between these concepts, save that Quantity is a pure determination of thought, while Matter is the same as existing externally. — The Ego, too, is determined as pure Quantity, since jt is an absolute alienation, an infinite removal or universal
repulsion towards the negative freedom of Being-for-Self, though it remains absolutely simple continuity — the continuity of uni¬ versality or Being-with-Self, which is not interrupted by the infinite variety of limits, the content of sensations, intuitions, and so forth. — Those who refuse to take plurality as simple unity, and want an image of this unity, apart from the concept (namely, that each of the Many is the same as every other, that is, One of the Many, — since we are not here speaking of many further determined, of green, red, and so on, but of the Many considered in itself), will find plenty of examples among those static existences which offer as present the deduced concept of Quantity in simple intuition.
It is the nature of Quantity to be this simple unity of Dis¬ creteness and Continuity'; and to this appertains the conflict or Antinomy of the infinite divisibility of Space, of Time, of Matter, and so on.
This antinomy consists solely in the necessity of asserting Discreteness as much as Continuity. The one-sided assertion of Discreteness gives an infinite or absolute division (and thus something indivisible) for principle ; and the one-sided assertion of Continuity, infinite divisibility.
Kant’s Critique of Pure Reason sets up, as is well known, four (cosmological) Antinomies, the second of which deals with the opposition which is constituted by the moments of Quantity.
These Kantian antinomies still remain an important part of the critical philosophy; they, principally, effected the fall of the previous metaphysics, and may be looked on as a chief transition to modern philosophy ; for they in particular assisted to produce a conviction of the invalidity of the categories of finitude by examining their content ; — and this is a more correct method than that formal method of a subjective idealism according to which their only fault is supposed to be that they are subjective, and not that which they are in themselves. But, great as are its merits, this exposition is very incomplete, being in part self-hampered and cross-grained, and in part faulty in view of its result which assumes that cognition has no other form of thought than finite categories. — In both respecfg these
i don’t have a funny review. i really liked the film. it made me sad and was very good.
antinomies deserve a more exact criticism, as well investigating their standpoint and method more closely as liberating the main and essential point from the useless mould into which it has been forced.
First I remark that Kant tried to give an appearance of completeness to his four cosmological antinomies by his method of classification, which he borrowed from his scheme of cate¬ gories. A deeper insight into the antinomous or, rather, into the dialectic nature of Reason shows, however, that every con¬ cept is a unity of opposite moments, which could therefore be asserted in the shape of an antinomy. Thus, Becoming, Deter¬ minate Being, and so on, and other concepts, could each furnish its particular antinomy, and as many antinomies could be set up as concepts were yielded. — The old scepticism did not shrink from the labour of demonstrating this contradic¬ tion or antinomy in every concept which it found in the sciences.
Next, Kant took the antinomy not in the concepts themselves, but in the already concrete form of cosmological determina¬ tions. In order to have a pure antinomy which might be dealt with in its simple concept, it was necessary not to take the determinations of thought in their application to and admixture with the ideas of World, Space, Time, Matter, and so on, but to consider them purely for themselves without this con¬ crete material, which here has no force nor avail; for these determinations alone constitute the essence and base of the antinomies.
Kant offers this concept of the antinomies, that they are no tricks of sophistry, but contradictions which Reason must necessarily hit on (as Kant puts it) : a view which is important. —“The natural appearance of the antinomies no longer actively deludes Reason when it perceives its base; yet still Reason is deceived by it.” — Naturally; for the critical solution by means of the so-called transcendental ideality of the world of per¬ ception has no other result than this, that it makes the so-called conflict into something subjective; and there, of course, it remains appearance, that is, it remains unresolved, as much as before. The true solution can only be this, that two deter¬ minations, being contradictory, and yet necessary to the same concept, cannot be valid each by itself in its one-sidedness, but
have their truth only in their transcendence, in the unity of their concept.
More closely considered, the Kantian antinomies contain no more than the simple categorical assertion of each one of the two opposed moments of a determination, apart from the other. But while this is done, the simple categorical or rather assertorical statement is enveloped in a scaffolding of reasoning, awry and crooked, intended to produce a show of proofs and to hide and disguise its merely assertorical character. This will become clear on closer examination.
The antinomy proper to this place is that of the so-called infinite divisibility of matter, and is based on the opposition between the moments of continuity and discreteness contained in the concept of Quantity.
“Every composite substance in the world consists of simple parts, and there exists everywhere nothing but what is simple or is composed of simple parts.”
Here to the simple, the atom, is opposed the composite, which as compared with the static or continuous is a rudi¬ mentary determination. — The substratum given to these ab¬ stractions (namely, substances in the world) means no more than things as perceived by the senses, and is irrelevant to the antinomy; Space or Time might equally well have been taken. — Since now the thesis deals only with compositeness instead of continuity, it is really an analytical or tautological proposi¬ tion. That the composite is not One in itself, but is only something put together externally and consisting of Other, is its immediate determination. But the Other of the composite is the simple. It is therefore a tautology to say that the com¬ posite consists of the simple. — If it is asked of what something consists, it is demanded that something other be indicated, the compounding of which constitutes this something. If ink is to be made up just of ink, then the meaning of the question which asks of what other bodies it consists is missed and, not being answered, merely repeats itself. It is then a further question whether the object of enquiry is to consist of something or not. But the composite is just that which is supposed to be a con¬ glomeration and to consist of others. — If the simple (which is allowed to be Other to the composite) is taken as njcrely
relatively simple, but again composite in itself, then the question still remains. Imagination has before it only this or that concrete composite, of which this or that Something might be indicated as its simple unit, though composite in itself. But here we are speaking of the composite as such.
mister numbers meets god and calls him five and dies but is reborn i suppose. the moneymen just want him for his numbers, the godmen just want him for his numbers, but he does not want the numbers. giving up and forgetting and moving on is the point i suppose.
As regards Kant’s proof of the Thesis, it — like all Kant’s proofs of the remaining antinomous propositions — makes the detour (which will prove highly superfluous) of being apagogic.
“Assume” (he begins) “that the composite substances do not consist of simple parts ; then, if all composition were mentally removed, there would be no composite part; now we have assumed that there are no simple parts; there is therefore no simple part — and hence nothing at all — left; hence no substance is given us.” —
This conclusion is quite correct: if there is only composite and we imagine everything composite removed, then nothing remains; we may grant this, but this tautological superfluity might be spared, and the proof might begin with what follows, namely : —
“Either all composition cannot be mentally removed, or, it being removed, something subsisting without composition — namely, the simple — must remain.”
“But in the first case again the composite would not consist of substances (for with these compositeness is only a contingent relation of the substances,! and they must be capable of existing as persistent beings without it). — But this case is contradictory to the hypothesis, and thereiore only the second remains: namely, that all composite substance consists of simple parts.”
The reason which is the main point, compared with which all that has been said is quite superfluous, is relegated to a parenthesis. The dilemma is this : — That which persists is either the composite or else the simple. If the first (the composite) were that which persists, then that which persists is not the substances, for in these compositeness is a contingent relation; but the substances are that which persists, therefore that which persists is the simple.
It is evident that the apagogic detour might be avoided, and
' Here redundance of language it added to redundance in proof, — “for with these (to wit, the substances) compositeness is only a contingent relation of Its
that to the Thesis (“Composite substance consists of simple parts”) this reason might immediately be added as proof, namely that compositeness is merely a contingent relation of the substances, external to and not concerned with them. — If compositeness is in truth contingent, then the essence is of course the simple. But this contingency, which is the sole point at issue, is not proved but simply assumed (in parenthesis) as though it were obvious or beside the point. It is, of course, self-evident that compositeness is the determination of con¬ tingency and externality; but, if we were only about to deal with a contingent juxtaposition instead of continuity, it was not worth while to set up an antinomy about it, or, rather, it was impossible ; in that case it is mere tautology (as was said) to assert that the parts are simple.
Thus in the apagogic detour we see that the very assertion which is to be its result already occurs. Thus the proof might more briefly be put as follows : —
but “giving up” is just change with a certain context, a spin of your own definition. it could just as soon be labelled “changing focus” or “letting go” or any number of empty platitudes. mister numbers seemed to dislike the obsession, even if it was the only thing that gave him purpose. is he better off without the numbers in his head? who’s to say. all answers converge on “you can’t know, and that’s the point”.
Let it be assumed that substances do not consist of simple parts, but are composite. Now all composition can be thought away (since it is a merely contingent relation) ; it being removed, therefore, no substances remain unless they consist of simple parts. But we must have substances, since we assumed their existence ; everything must not vanish, something must remain ; for we have presupposed that some such persistent entity (which we called substance) exists. Therefore this something must be simple.
To complete matters, we must consider the conclusion, which runs as follows : —
“Hence it follows immediately that all the things in the world are simple essences, that compositeness is a condition merely external to them, and that reason must consider the elementary substances to be simple essences.”
Here the external, that is, the contingent, nature of com¬ positeness is cited as a consequence, after having already been introduced parenthetically and used in the proof.
Kant protests much that he is not looking for sophisms in the conflicting propositions of the antinomy, in order to effect (as the phrase is) an advocate’s proof. Indeed, this proof is not really to be accused of sophistry so much as of a forced and useless tortuosity, which serves only to produce the outer form
of a proof, not allowing it to be perfectly transparent that that which should stand out as conclusion is the (parenthetic) hinge of the proof, and that there is no proof but only an assumption.
“No composite thing in the world consists of simple parts, and nothing simple anywhere exists in it.”
The proof again has the apagogic turn, and in another manner is as faulty as the previous.
“Assume,” it runs, “that a composite thing as substance consists of simple parts. Now all external relation (and therefore also the composition of substances) is possible only in space, and therefore the space which it occupies must consist of as many parts as those of which the composite thing consists. Now space does not consist of simple parts, but of spaces. Therefore each part of the composite thing must occupy one space.”
“But the absolutely first parts of every composite thing are simple.”
rating: happy recovery brain injury man with freshly baked goods
“Now every reality which occupies a space comprehends a manifold of mutual externalities ; it is therefore composed (of substances), and thus the simple would be a substantial com¬ posite, which is self-contradictory.”
This proof may be called a perfect nest (to use an expression which is elsewhere employed by Kant) of faulty procedure.
First, the apagogic turn is a baseless fabric. The assumption that whatever is substantial is spatial, while space does not consist of simple parts, is a direct assertion which is made the immediate foundation of that which is to be proved ; after this there is no need for further proof.
Next, this apagogic proof begins with the proposition “that all composition of substances is an external relation,” but, strangely enough, immediately forgets it. For the further con¬ clusion is made that composition can take place only in space, that space does not consist of simple parts, and that therefore the reality occupying a space is composite. If once composite¬ ness has been taken as an external relation, then spatiality itself (since composition is supposed to be possible in it alone) is for that very reason a relation external to the substances, which does sot concern them nor touch their nature any more than
does all the rest that can be inferred from the determination of spatiality. For this very reason the substances should not be placed in space.
Further, it is assumed that the space into which the sub¬ stances are here transplanted does not consist of simple parts ; since it is an intuition (that is, according to Kant’s determina¬ tion, a sensuous representation) which can be given only by one single object, and is not a so-called discursive concept. — This distinction of Kant’s between intuition and concept has, of course, led to the abuse of the former, and, in order to save the labour of achieving the latter, its value and sphere have been extended to all cognition. What is relevant here is this, that space, like intuition itself, must be conceived if we would have any concept at all. And thus the question would arise whether space, though simple continuity for intuition, might not conceptually have to be taken as composed of simple parts, since otherwise space would be affected by that same antinomy into which substance only was transplanted. Indeed, if the antinomy is taken in the abstract, it extends to all Quantity (as was said above) and therefore also to Space and Time.
Now in the proof it is assumed that space does not consist of simple parts ; and this should have been sufficient reason for not transplanting the simple into this element, which is inade¬ quate to the determination of the simple. — Here also the con¬ tinuity of space comes into collision with compositeness ; these two are confused, and the former is inserted in the place of the latter (which results in a Quaternio Urminorum in the con¬ clusion). Kant expressly determines space as being one, its parts being based only on limitations, so that they do not precede space, sole and all-comprehending, as its component parts from which it could be put together. ( Critique of Pure Reason, Second Ed., p. 39.) Here continuity is predicated of space, very justly and definitely, in opposition to its compositeness out of parts. In the proof, on the other hand, the transplanting of the sub¬ stances into space is to involve “a manifold of mutual exter¬ nalities,” which, further, is “therefore composite.” Yet we have quoted what shows that the manner in which a manifold exists in space is expressly designed to exclude compositeness and parts which precede its unity.
In the Note to the proof of the Antithesis, the other Junda-
mental notion of the critical philosophy is further expressly adduced, namely, that we have a concept of bodies only as phenomena, which, as such, necessarily presuppose space as the condition of the possibility of all external manifestation. If by substances mere bodies are meant, such as are seen, felt, tasted, and so forth, then there is really no question of their concep¬ tual meaning : sense-perception only is in question. The proof of the Antithesis might then be put in brief thus : — All visual, sensory, and other experience shows us only what is composite; the best microscopes and the keenest knives have never yet allowed us to hit on anything simple. Therefore Reason, too, should not want to hit on anything simple.
And thus when now we regard more closely this opposition of Thesis and Antithesis, liberating their proofs of all idle superfluity and tortuousness, we find that the proof of the Antithesis, by transplanting the substances into space, contains a dogmatic assumption of Continuity, and that the proof of the Thesis, by assuming Compositeness as the class of relation sub¬ sisted between the substances, contains the dogmatic assump¬ tion of the contingency of this relation — that is, the assumption that the substances are absolute Ones. Thus the whole antinomy reduces itself to the separation and direct assertion of the two moments of Quantity as being absolutely separate. Looked at from the point of view of mere discreteness, substance, matter, space and time, and so on, are absolutely divided, and the One is their principle. From the point of view of continuity, this One is merely suspended: division remains divisibility, the possibility of dividing remains as possibility, without ever actually reaching the atom. Now even if we do not move beyond the determination implied in what has been said about these contradictions, still continuity itself contains the moment of the atom, since continuity exists simply as the possibility of division ; just as accomplished division, or discreteness, cancels all distinction between the Ones — for each simple One is what every other is, — and for that very reason contains their equality and therefore their continuity. Each of the two opposed sides contains its other in itself, and neither can be thought of with¬ out the other; and thus it follows that, taken alone, neither determination has truth, but only their unity. This is the true dialectic consideration of them, and the true result.
The dialectic examples of the old Eleatic school, especially those which deal with motion, are incomparably deeper and richer in meaning than the antinomy of Kant which we have considered. They, too, are based on the concept of Quantity, and in it have their solution. To consider them here would lead too far ; they concern the concepts of Space and Time, and can be dealt with in connexion with these subjects and with the history of philosophy. — They do the highest honour to the reason of their inventors; their result is the pure Being of Parmenides, since they demonstrate the solution into itself of all determinate being : in themselves, therefore, they are the Flux of Herakleitos. They therefore deserve a fuller considera¬ tion than the ordinary explanation that they are just sophisms ; — an assertion which clings to empirical perception, following that method of Diogenes (so convincing to common sense), who, when a dialectician demonstrated the contradiction con¬ tained in motion, is said to have put no further strain on his reason, but, by mutely walking up and down, to have referred to the evidence of the eyes; — an assertion and a refutation which of course it is easier to make than to enter upon thought, to seize the confusions into which thought, quite unforced, leads when it formulates itself in ordinary consciousness, and to solve them by means of thought.
The solution of these dialectic formations which Aristotle effects is highly to be praised : it is implied in his truly specu¬ lative concepts of Space, Time, and Motion. Infinite divisibility (which, being imagined as actually completed, is equivalent to infinite division, or the atoms), as being the principle on which are based the most famous of these proofs, is by him opposed to Continuity (of Time as well as of Space), in such a manner that infinite — that is, abstract — plurality is contained in continuity only in itself, or potentially. The actual, as opposed to abstract plurality (or to abstract continuity), is their concrete form, it is* Time and Space itself; and these in turn are opposed to Motion and Matter. The abstract exists only in itself, or potentially ; it exists only as a moment of the Real. Bayle in his Dictionnaire (Article, Zenon) considers Aristotle’s solution of Zeno’s dialectic “ pitoyable ” : he does not understand the meaning of the potentially infinite divisibility of matter. He replies that if matter is infinitely divisible, it actually contains
an infinite number of parts", so that this is not an infinite en puissance but an infinite really and actually existing. — No: divisibility itself is only a potentiality, not an existence of parts ; and in any case, in Continuity, plurality is posited only as a moment already transcended. — Keen understanding — though in this, too, probably Aristotle has not been excelled — is not sufficient to seize and judge his speculative concepts; nor is the clumsy method of sensuous representation which we cited sufficient to refute such arguments as Zeno used. Such an understanding makes the mistake of taking these objects of thought or abstraction, such as an infinite number of parts, for Something, for true and actual ; and this sensuous conscious¬ ness will never let itself be raised from the sphere of the empirical to that of Thought.
Kant’s solution of the antinomy, too, consists only in demand¬ ing that Reason must not rise above sense-perception, but must take the phenomenon as it is. This solution leaves on one side the content of the antinomy ; it does not reach the nature of the concept of its determinations, each of which, isolated by itself, is null, being no more than a transition to its Other, while Quantity is their unity and therefore their truth.
I. Quantity contains the two moments of Continuity and Discreteness. Each is its determination, and it must therefore be posited in each. — Primarily it is their immediate unity, that is, primarily it is posited only in one of its determinations, namely, continuity ; and thus is Continuous Magnitude.
Or again, continuity is one of the moments of Quantity which requires the other moment, discreteness, to complete it. But still Quantity is concrete unity only in so far as it is the unity of distinct moments. The moments therefore must be taken as distinct, but must not be resolved back into Attraction and Repulsion ; they must be taken in their truth, as each forming the whole in its unity with the other. Continuity is only coherent and homogeneous unity as unity of discrete elements.
and, posited thus, it is no longer mere moment but complete Quantity: this is Continuous Magnitude.
2. Immediate quantity is Continuous Magnitude. But in fact Quantity is not immediate; immediacy is a determinate¬ ness, and Quantity itself is this immediacy transcended. It must therefore be posited in the determinateness immanent in it, and this is the One. Quantity is Discrete Magnitude.
Discreteness, like continuity, is a moment of Quantity, but is also the whole of Quantity just because it is a moment of this whole, and thus, as distinct from it, does not relinquish its unity with the other moment. — Quantity is externality in itself, and Continuous Magnitude is this externality as pro¬ pagating itself without negation, as a context which remains at one in itself. Discrete Magnitude is this externality as non- continuous or interrupted. But though we thus have a multi¬ tude of Ones, this is not the multitude of atoms and the void, or Repulsion in general, meeting us once again: Discrete Magnitude is Quantity; and for that very reason their dis¬ creteness is continuous. The continuity in discreteness consists in the fact that the Ones are equal to one another, or have the same unity. Discrete Magnitude, then, is the externality of much One posited as the same, and not of the many Ones in general ; it is posited as the Many of one unity.
Ordinarily, when an image is formed of Continuous and of Discrete Magnitude, it is overlooked that each of these mag¬ nitudes has both the moments, continuity and discreteness, and that their difference is only constituted by which of the two moments is posited determinateness, and which that which only is in itself. Space, Time, Matter, and so on, are fixed mag¬ nitudes because each is a repulsion from itself, where there is a stream of extrogression without any transition or relation to what is qualitatively Other. They have the absolute poten¬ tiality of the One being posited relatively to them, and not merely as the empty potentiality of a mere otherness (as we say that a tree might occupy the space occupied by a stone) : they contain in themselves the principle of the One, it is one of the determinations which constitute them.
[sic; i’ll leave figuring out who wrote this one as an exercise to the reader. make sure to write up a proof in lean
Conversely, continuity must not be overlooked in Discrete Magnitude: this moment, as has been shown, is the One as unity.
Continuous and Discrete Magnitude may be considered as species of Quantity, but only in so far as magnitude is posited not under any external determinateness, but solely under the determinatenesses of its own moments ; the ordinary transition from genus to species allows external determinations to be applied to these according to some external principle of classi¬ fication. And so far Continuous and Discrete Magnitude are not yet Quanta: they are Quantity itself in each one of its two forms. If they are called magnitude, this is because they have this in common with Quantum, to be a determinateness applied to Quantity.
Discrete Magnitude, first, has the One for principle ; secondly, it is a plurality of Ones ; thirdly, it is, essentially, static ; it is One, but One transcended, or as unity; it is self-continuation as such in the discreteness of the Ones. It is therefore posited as one magnitude, and its determinateness is the One, which, in this positedness and determinate being is exclusive One, or Limit, applied to unity. Discrete Magnitude as such is not to be limited immediately; but, as distinct from Continuous Magnitude, it exists as a Determinate Being and a Something whose determinateness is One, and, as being in a Determinate Being, is also first negation and Limit.
This Limit is referred to unity and applied to it as negation, and, besides this, as One is also referred to itself; it is thus including and comprehending Limit. Here the Limit is not distinguished from the Something of its Determinate Being, but, as One, is immediately this negative point itself. But the Being which here is limited exists essentially as continuity, by virtue of which it passes beyond the Limit and beyond this One, and is indifferent to them. Real discrete Quantity is therefore one Quantity, or Quantum, — Quantity as Determinate Being and Something.
Since the One, which is Limit, comprehends in itself the many Ones of discrete Quantity, it in so doing also posits them as transcended in itself. It is Limit as applied to continuity in general as such, and hence the distinction between Continuous and Discrete Magnitude is here immaterial ; or, rather, it is Limit to the continuity of the one as much as of the other: both pass over into the state of Quantum.
Quantum — which, first, means Quantity having any deter¬ minateness or limit at all — is, in its complete determinateness, Number. It is divided into
secondly [a) Extensive Quantum, where the Limit is the Barrier of a plurality there existing; ( b ) Intensive Quantum or Degree, the plurality having passed over into bcing-for-self. This Barrier being for itself, and therefore as Limit indifferent, is equally immediately outside itself and has its determinate¬ ness in an Other. Thus Quantum is a posited contradiction, it is simply determined for itself and also has its determinate¬ ness outside itself and thus refers beyond itself; it therefore
thirdly passes over, as that which is itself posited as external to itself, into Quantitative Infinity.
Quantity is Quantum ; it has a Limit, whether it be taken as continuous or as discrete magnitude. The distinction between these two species is here of no importance.
As transcended Being-for-Self, Quantity in itself already is indifferent to its Limit. For that very reason Limit (or the fact that it is a Quantum) is not indifferent to it ; for it contains the One, or absolute determinateness, in itself as its own moment, which thus posited in its continuity or oneness is its Limit, but yet remains that One which it has become.
This One is thus the principle of Quantum; but it is the One of Quantity. Hence, first, it is continuous, or Unity; secondly, it is discrete, plurality of Ones which is in itself (as in continuous magnitude) or posited (as in discrete mag¬ nitude) : these Ones are equal to one another, they have this continuity, this same unity. Thirdly, this One is also negation of the many Ones as simple Limit, an exclusion of otherness.
or a determination of itself in opposition to other Quanta. One is thus Limit (a) referring to itself, (j8) inclusive, and (y) ex¬ clusive of Other.
Quantum completely posited in these determinations is Number. Complete positedness is implied by the existence of the Limit as plurality and its consequent distinctness from unity Hence Number appears as discrete magnitude, but in unity it also has continuity. Hence too it is Quantum in com¬ plete .determinateness : for in it Limit is determinate plurality, having for principle One, the absolutely determinate. Con¬ tinuity, where One is only in itself or as transcended (posited as unity) — is the form of indeterminateness.
Quantum, merely as such, is limited generally; its limit is its abstract and simple determinateness. But since it is Number, the Limit is posited as being manifold within itself. It contains the many Ones which constitute its Determinate Being, but not in a merely indeterminate manner: the determinateness of the Limit falls within it. The Limit excludes other Deter¬ minate Being, that is, other Manies : the Ones included in it are a determinate heap, an Amount. This is discreteness as it is in Number, and hence its other is Unity, the continuity of this. Amount and Unit constitute the moments of Number.
With regard to Amount, it must further be considered in what manner the many Ones of which it consists are in the Limit; it is correct to say of Amount that it consists of the Many, for in it the Ones do not exist as transcended, but in it they are , posited only with the excluding Limit, to which they are indifferent. The Limit is not, however, indifferent to them. The relation of Determinate Bang to Limit (as was seen when that category was treated) was such that Determinate Being, as the affirmative, remained on this side of its Limit, while the Limit (or negation) remained outside, at the edge; similarly, with the many Ones, the abrupt exclusion of other Ones appears as a determination falling without the included Ones. But it was seen above that Determinate Being is pene¬ trated by Limit and is coextensive with it, and that thus some¬ thing is inherently limited, that is, finite. — Thus Number — which is Quantitative— is imagined in such a way that with one hundred, for instance, the hundredth One alone shall limit the Many in such a manner as to make them one hundred-
This is correct ; yet among the hundred Ones none is preferred individually, since all are equal : each equally is the hundredth ; they all equally belong to the Limit which makes the number one hundred, and, for the determinateness of the number, not one can be spared. Thus the others do not constitute relatively to the hundredth One a Determinate Being external to the limit, or distinct from it within it. Amount then is not a plurality as opposed to the inclusive and limiting One, but rather itself constitutes the limitation which is a determinate Quantum : the Many constitute one number, One two, One ten, One hundred, and so forth.
Now the limiting One is determinateness against Other, the distinguishing of one number from another. But this dis¬ tinguishing does not become qualitative determinateness; it remains quantitative and appertains only to external com¬ paring reflection: Number, as One, has returned upon itself, and at this resting-place it remains indifferent to Others. This indifference of Number to Others is its essential determination : by virtue of it, Number is determinate-in-itself, but also in itself external. — It is thus a numerical One, as absolutely determinate ; it has the form of simple immediacy, and there¬ fore relation to Other is quite external to it. And as being One which is Number, it further has determinateness (in so far as it is relation to Other) as its moments in itself, in its distinction between Unit and Amount ; and Amount itself is plurality of Ones, that is, in itself it is this absolute externality. — This contradiction of Number, or of Quantum in general, in itself, is the quality of Quantum : this contradiction develops itself in the further determinations of the latter.
Spatial and numerical magnitude are generally considered as two species, each being a determinate magnitude as much as the other; and it is thought that, the distinction being only in the different determinations of continuity and discreteness, they rank equally as Quantum. Geometry, in spatial magni¬ tude, generally has continuous magnitude for object; and arithmetic, in numerical, has discrete magnitude for object. But, the objects being thus different, they are not limited or
determined in the same manner or with the same completeness. Spatial magnitude is merely limited in the abstract ; if it must be considered as a definitely determinate Quantum, Number is required. Geometry as such does not measure spatial figures — it is not the art of measuring: it only compares. In its definitions too the determinations partly are derived from equality of sides and angles, or from equidistance. Thus the circle, since it is based solely upon the equidistance of all possible points in it from a centre, can be determined with¬ out Number. These determinations, based on equality and inequality, are true geometry. But they are not sufficient, and others, such as triangle or quadrilateral, require Number, which in its principle, the One, contains determinedness-for- self, not determinedness by the aid of an Other, hence not by comparison. It is true that, in the point, spatial magnitude has a determinateness corresponding to the One; but the point, in so far as it passes out of itself, becomes an Other, becomes the line ; and, being essentially the One of Space, it becomes, when related, a continuity, in which its puncticity (or determinedness-for-self, or the One) is transcended. In so far as determinedness-for-self is to preserve itself in self-exter¬ nality, the line must be imagined as a multitude of Ones, and the Limit must contain the determination of many Ones ; that is, the magnitude of the line — like other spatial determinations — must be taken as Number.
Arithmetic contemplates Number and its figures ; or, rather, it operates with them and does not contemplate. For Number is indifferent determinateness and inert; it must be actuated and brought into relation from without. The different methods of relation are the species of calculation. In arithmetic they are enumerated in series, and it is evident that they are mutually dependent. But arithmetic does not give prominence to the thread which guides their progress. However, the systematic arrangement which the exposition of these elements in the text-books justly claims, results easily from the conceptual determination of Number itself. We will here briefly notice these cardinal determinations.