This week minfeub watched Alien 1979. Rachel picked it because it’s one of her favorite horror movies
which, in so far as it had a limit as such, would be just a multitude, that is, a something limited but limited indeter¬ minately; in order to make it into a determinate Quantum, the Many must be subsumed into One, by which process they are posited as identical with the limit. Continuous and discrete magnitude, taken as Quantum generally, have each posited in it only one of the sides by which it is completely determined and made into Number. The latter is immediate Extensive Quantum, — simple determinateness, existing essentially as amount , though only as an amount of one and the same unit, being distinguished from Number only in this, that in the latter determinateness is expressly posited as plurality.
2. However, when the magnitude of something is determined by number, no contrast with another magnitude is needed: the determinateness of this magnitude does not depend upon itself and another magnitude, for the determinateness of mag¬ nitude in general is self-determinate, indifferent and simply self-related Limit; which, in Number, is posited as enclosed in the self-existing One, externality and relation to Other being within itself. This Many of the Limit, further, (like the Many in general) is not differentiated within itself, but continuous; each of the Many is what every other is; its determinateness as such does not consist in its quality of mutual externality of parts, or discreteness. Thus this Many for itself collapses into its continuity and becomes simple unity. — Amount is only a moment of Number, but does not constitute the determinate¬ ness of Number as a multitude of numerical Ones : the Ones, as indifferent and self-external, are transcended when Number has returned upon itself. Externality, which constituted the Ones of plurality, vanishes in the One as self-relation of Number.
Quantum, as extensive, had its existing determinateness as self-external Amount: its limit thus passes over into simple determinateness. In this simple determination of the limit. Quantum is intensive magnitude; and the limit or deter¬ minateness which is identical with Quantum is now also posited as simple — it is Degree.
Degree thus is determinate magnitude or Quantum, but it is not also a multitude or multiplicity within itself; it is only a “more”; and the quality of “more” is the manv taken
together in the simple determination. Determinate Being which has returned to Being-for-Self. Its determinateness re¬ quires a number (which is the complete determinedness of Quantum) to express it; but the number is not Amount but, simply, a Degree. If we speak of ten or twenty degrees, then the Quantum which has so many degrees is the tenth or twentieth degree, and not the amount and sum of them; for, if so, it would be extensive; but in fact it is only one — the tenth or twentieth — degree. It contains the determinateness implied in the amount ten or twenty, but does not contain it as many; it is Number as transcended amount or simple determinateness.
3. In Number, Quantum is posited in its complete deter¬ minateness; but as Intensive Quantum, as in the Being-for- Self of Number, it is posited as it is according to its notion, or in itself. For the form of self-relation which it has as Degree is also its self-externality. As Extensive Quantum, Number is numerical plurality, and thus contains externality within itself. The latter, as being Many in general, collapses into undifferen- tiatedness and transcends itself in the One of Number, which is its self-relation. Now Quantum has determinateness when taken as Amount : as shown above, it contains amount though amount is not posited in it. And so Degree, as being simple in itself, no longer has this external otherness within itself; it has it outside itself, and it stands related to it as to its determi¬ nateness. A plurality external to it constitutes the determinate¬ ness of the simple limit, which Degree is for itself. If, therefore, Amount (so far as, in Extensive Quantum, it existed within Number) there transcended itself, it passes into being posited outside Number. Number being posited as One, as intro- reflected self-relation, excludes from itself the indifference and externality inherent in Amount, and is self-related because through itself it is related to something external.
Herein Quantum has the reality proper to its notion. The indifference of determinateness constitutes its Quality; that is, determinateness which in itself exists as self-external deter¬ minateness. — Degree accordingly is a simple determination of magnitude among several such intensities, which are different, each being a simple self-relation but also being in an essential relation to the others, so that each has its determinateness in
this continuity with the others. Since, then, Degree is related to its Other through itself, the ascent or descent in a scale of degrees becomes a steady progress or flow, which is an unin¬ terrupted and indivisible mutation; not one of the several entities which are distinguished therein is separated from the others, but each has its determinateness only in them. As self- related determination of magnitude each degree is indifferent to the others ; but also it is implicitly related to this externality, and is what it is only through it ; in a word, its self-relation is the non-indifferent relation to the external, and in this fact consists the quality of the degree.
Degree is not self-external within itself. But it is not indeter¬ minate One, or that general principle of Number which is no amount except that negative amount implied in the fact that it is not amount. First, Intensive Magnitude is a simple One of the heap ; there are several degrees ; but they are determined neither as simple One nor as heap, but only in the relation of this self-externality, or in the identity of One and plurality. So that, if members of a plurality, as such, are outside the simple degree, its determinateness consists in its relation to them: it thus contains Amount. Twenty, as Extensive Mag¬ nitude, also contains twenty discrete Ones ; on the other hand, the determinate degree contains them as a continuity; and this determinate plurality is this continuity simply. It is the twentieth degree, and is such only by virtue of this amount, which as such is external to it.
The determinateness of Intensive Magnitude must therefore be considered from two aspects. It is determined through other intensive Quanta, and it is continuous with its otherness, in such a manner that its determinateness consists of its relation to this. First, then, in so far as it is simple determinateness, it is determinate relatively to other degrees ; it excludes them, and this exclusion constitutes its determinateness. Secondly, however, it is determinate in itself : it is determinate in amount, as in “its’* amount, not in amount as excluded nor as taken as the amount of other degrees. The twentieth degree contains twenty in itself; it is determinate, not only as distinguished
That was a nice film, it was able to stay engaging through it’s full two hours which i felt some films in the past struggled with, which was nice. The design of the ship was wonderful, as was the design of the alien ship and the alien itself for its first metamorphical stages (i’m not a huge fan of the xenomorph design [im like 90% sure that this it was a xenomorph] but that’s a personal digression i just have with seeing it over the years). The latter half felt a little bit too epileptic for my tastes, but otherwise it was fine. The characters were engaging, i liked the spheres in the robot guy’s throat, and the false ending was pleasant and engaging.
from the nineteenth, twenty-first, or other degrees ; but, further, its determinateness is its amount. But, in so far as the amount is its own — and determinateness also essentially exists as amount — degree is Extensive Quantum.
Thus Extensive and Intensive Magnitude are one and the same determinateness of Quantum; they only differ in that one has amount within itself, and the other without. Extensive Magnitude passes over into Intensive Magnitude because the Many of Extensive Magnitude in itself collapses into the unity to which the Many is external. But conversely this simple result has its determinateness only in amount, and in fact in “its” amount; it is indifferent to intensities otherwise determined, and, therefore, has the externality of amount in itself; and thus Intensive Magnitude equally essentially is Extensive Magnitude.
Having reached this identity, we have reached the qualita¬ tive Something; for this identity is unity self-related by virtue of the negation of its distinctions; now the latter constitute the existing rieterminateness of magnitude : this negative identity thus is Something, Something moreover which is in¬ different to its quantitative determinateness. Something is a Quantum, but now qualitative Determinate Being, as it is in itself, is posited as indifferent towards it. Quantum, Number as such, and so forth, could be spoken of without mention of a Something such as should be their substratum. Now, however, Something stands opposed to these its determinations as exist¬ ing for itself; it has become mediated with itself through their negation, and, in having a Quantum, is opposed to them as that same Something which is defined as having Extensive and Intensive Quantum. Its one determinateness proper to it as Quantum is posited in the distinct moments of Unit and Amount. Now not only is this determinateness one and the same in itself, but, when it is posited in these distinctions as Extensive and Intensive Quantum, there is a return to that unity which, as negative, is the Something which is posited as indifferent to it.
In common imagination Extensive and Intensive Quantum generally are distinguished as different kinds of magnitude, as
though some objects had only intensive and others only ex¬ tensive magnitude. To this has been added the view proper to a philosophy of nature, where the many, or that which is extensive (as happens, for instance, in the fundamental deter¬ mination of matter — namely, that it occupies space — and in other concepts), is changed into something intensive in this sense, that the true determination was supposed to be the equation of the intensive with the dynamic; it was thought that, for instance, density (or the specific occupation of space) must be taken essentially as a certain degree of the space- occupying force of matter, and not as a certain multitude and amount of material parts in a quantum of space.
Here two determinations must be distinguished. In the so- called transvaluation of the mechanistic into the dynamic view, there occurs the concept of independent parts existing apart from one another (the nexus which makes them a whole being merely external), and the distinct concept of force. That which, as occupying space, was considered merely as a multitude of mutually external atoms, is now regarded as the manifestation of a simple latent force. — But these relations of Whole and Parts, of Force and its Manifestation, which are here opposed to one another, do not belong to this place, and will be con¬ sidered later. But this much may be said, that the relation of Force and its Manifestation, which corresponds to intensity, is more valid than the relation of Whole and Parts ; but that in spite of this force is equally as one-sided as intensity, and that manifestation (the externality of extensiveness) is equally in¬ separable from force, so that one and the same content is found alike in both forms, in the intensive as well as in the extensive.
The other determinateness which here occurs is quantitative determinateness as such, by which is meant a determinateness which is transcended as Extensive Quantum and is changed into Degree, which is supposed to be the valid determination. But it has been shown that Degree too contains the former, so that one form is essential to the other, and each Determinate Being represents its determination of magnitude as Extensive and as Intensive Quantum equally.
Everything, so far as it appears as a determinate magnitude, will serve as example for this. Number itself necessarily bears
immediately in it this double form. It is amount, and in so far it is extensive magnitude; but also it is One, one Ten, one Hundred, and in so far it stands at the point of transition to intensive magnitude, since in this unity the multiple collapses into simple. One is extensive magnitude in itself, and may be imagined as any desired amount of parts. Thus the tenth or hundredth part is this simple intensity which has its deter¬ minateness in the Many which fall outside it, that is, in extensity. A number is ten or one hundred, and also is the tenth or one hundredth in the numerical system; and each determinateness is identical with the other.
In the circle the One is called Degree, because any part of a circle essentially has its determinateness in the Many external to it, and is determined as merely one of a limited amount of such Ones. As a mere spatial magnitude the degree of the circle is a common number; regarded as degree it is an inten¬ sive magnitude, which only has meaning as determined by the amount of degrees into which the circle is divided; just as Number in general has its meaning only in the numerical series.
11/12, a rather satisfying film with good design
The magnitude of a more concrete object also has this twin aspect of extensive and intensive, and manifests this in the twin determinations of its Determinate Being, in one of which it appears as external, while in the other it appears as internal. Thus, for instance, a mass, as weight, has extensive magnitude, since it constitutes an amount of pounds, hundredweights, and so forth ; and intensive magnitude in so far as it exerts a certain pressure: the magnitude of pressure is simple, it is a degree which has its determinateness in the scale of degrees of pressure. Mass, as exerting pressure, appears as a Being- in- Self or Sub¬ ject, which has an intensive differentiation of magnitude. — Conversely, that which exerts this degree of pressure has the power to displace a certain amount of pounds (and so on), and therein lies the measure of its magnitude.
Again, heat has a degree, and this degree (whether the 10th, 20th, or any other) is a simple sensation, is subjective. But this degree equally exists as extensive magnitude, as the expansion of a liquid, of mercury in the thermometer, of air, of sound, and so on. A higher degree of temperature expresses itself as a longer column of mercury, or as a narrower sound-cylinder;
it heats a greater space in the same manner as a lesser degree heats a smaller space.
The higher note, as the more intensive, is also a greater multitude of vibrations ; and a louder note (to which we ascribe a higher degree) is audible in a greater space. — A more inten¬ sive colour will serve to stain a larger surface than a weaker colour can similarly stain; that which is brighter (again a different kind of intensity) is visible farther than the less bright; and so forth.
And in the spiritual sphere high intensity of character, of talent, or genius, has a similarly comprehensive being, extended action, and manifold contact. The profoundest concept also has the most universal meaning and application.
Kant has employed the application of the determinateness of Intensive Quantum to a metaphysical determination of the soul in a peculiar manner. In his critique of the metaphysical propositions about the soul (which he calls Paralogisms of Pure Reason), he comes to consider the deduction of the permanence of the soul from its simplicity. To this conclusion he objects in these terms ( Critique of Pure Reason , p. 414) : — “Although we grant that the Soul is of this simple nature, since it contains no manifold of separate entities, and therefore no extensive magnitude, yet we cannot deny to it, any more than to any other existent, intensive magnitude, that is, a degree of reality with regard to all its powers and in general with regard to everything that exists; this degree can decrease through an infinity of lesser degrees, so that this would-be substance can be changed into nothing by gradual relaxation ( remissio ) of its forces if not by division ; for consciousness itself always has a degree which can still be diminished; this therefore must apply also to the faculty of self-consciousness and to all the other faculties.” — In rational psychology — and this abstract metaphysic was such — the soul is considered not as Spirit but as a merely immediate Being, as a soul-thing. Kant thus has the right to apply the category of Quantum to it “as much as to any other existent ;” and, in so far as it is determined as simple, to apply the category of Intensive Quantum. It is true
that Being is a part of Spirit, but a Being of an intensity wholly different from that of Intensive Quantum ; an intensity rather where the form of mere immediate Being and every category of it have been transcended. Not only the removal of the category of Extensive Quantum should have been admitted: Quantum should have been removed altogether. But it is yet a further matter to understand how existence, consciousness, and finitude are in the eternal nature of Spirit and emerge from it, while it does not thereby become a thing.
The distinction between Extensive and Intensive Quantum is indifferent to the determinateness of Quantum as such. But, altogether, Quantum is determinateness posited as transcended, it is the indifferent limit and a determinateness which is also its own negation. This distinction is developed in Extensive Magnitude; Intensive Magnitude is the presence of this ex¬ ternality which Quantum is in itself. It is posited as its own contradiction in itself, where it is simple self-related deter¬ minateness and also that self-negation which has its deter¬ minateness not in itself but in another Quantum.
According to its Quality, therefore, a Quantum is posited in absolute continuity with its externality and otherness. Thus, it is not merely possible to pass beyond every determinateness of magnitude, and possible to alter it ; but it is posited that it must alter. The determination of magnitude continues itself into its otherness in this manner, that it has its being only in this continuity with an Other: it is a limit which is not, but becomes.
The One is infinite, or is self-related negation, and therefore self-repulsion. Quantum too is infinite, being posited as self- relating negativity ; it repels itself from itself. But it is a deter¬ minate One, a One which has passed over into Determinate Being and Limit : it is the self-repulsion of determinateness and not the creation of the self-equal (as happens in the repulsion of One) but of its otherness : it is now posited in itself to refer beyond itself and to become an Other. It consists in increase or diminution : it is in itself the externality of determinateness.
-recima from minfeub
Other, and this also is, primarily, Quantum ; but it is this as a limit which is not, but forces itself beyond itself. Thus the limit which arises again in this outward passage is simply one which again transcends itself and refen to yet another, and so on to infinity.
Quantum alters and becomes another Quantum ; the further determination of this alteration, that it proceeds to infinity, is implied in the fact that Quantum is posited as self-contra¬ dictory. — Quantum becomes an Other; but it continues itself into its otherness; the Other too is, therefore, a Quantum. But this latter is the Other not only of a Quantum, but of Quantum itself : it is the negative of Quantum taken as limited ; it is, therefore, its unlimitedness, or Infinity. Quantum is an Ought : it is determined for itself, and this is a part of it, and this very determinedness is, rather, a determinedness in an Other. And, inversely, it is transcended determinedness in an Other, it is indifferent self-subsistence.
Finitude and Infinity thus forthwith each receive a double, and in fact an opposite, meaning. Quantum is finite first because it is limited, and secondly because it refers beyond itself, and has its determinedness in an Other. And Quantum is infinite, first, in its non-limitedness, and, secondly, because, having returned upon itself, it is indifferent Being-for-Self. If, now, we compare these moments, we find that the determination of finitude in Quantum, that reference beyond itself to an Other which constitutes its determination, is likewise deter¬ mination of the infinite : the negation of the limit is a similar passing beyond determinateness, so that Quantum is to have its last determinateness in this negation, the infinite. The other moment of Infinity is that Being-for-Self which is indifferent to limit, but Quantum itself is limited in such a manner that relatively to its limit (and, therefore, to other Quanta and to its beyond) it is in itself indifferent. Finitude and Infinity (of the sort which tries to be separate from the finite — the v<3l. i
bad infinite) in Quantum each already contain the other for moment.
Qualitative and Quantitative Infinity are distinct because in the former the opposition between finite and infinite is qualitative, and the transition from the finite into the infinite, or the relation of either one to the other, subsists only in the “in-itself,” in their notion. Qualitative determinateness is immediate, and is related to otherness essentially as to another Being; and it is not posited as having its negation, its Other, in itself. But magnitude is, as such, transcended determinate¬ ness: it is posited as being both unequal and indifferent to itself, and, therefore, as being that which is alterable. Quali¬ tative Finitude and Infinity are, therefore, opposed absolutely, that is, abstractly ; their unity is the fundamental inner relation ; and, therefore, the finite continues itself into its Other only in itself and not in the infinite. But the quantitative finite refers itself to its infinite in the latter itself, claiming to have its absolute determinateness in it. This relation, then, is repre¬ sented by the Quantitatively Infinite Progress.
In general an infinite progress is the expression of a con¬ tradiction : here it expresses the contradiction contained in the quantitatively finite, or in Quantum in general. It is that reciprocal determination of finite and infinite which was con¬ sidered in the sphere of Quality, with this difference that, as has just been pointed out, in the quantitative the limit in itself refers and continues beyond itself, and that therefore, con¬ versely, the quantitatively infinite is posited as containing Quantum in itself, — for Quantum, in its self-externality, is also itself; its externality is part of its determination.
Now the Infinite Progress expresses but does not solve this contradiction; but, by reason of the merging of one deter¬ minateness into the other, it produces an apparent solution by uniting the two. As it is primarily posited, it makes the infinite the goal, which, however, is not reached : it is a per¬ petuated creation of it, while yet Quantum is never left behind and the infinite never becomes positive and present. It is part of the notion of Quantum to have a beyond. This beyond is,
first, the abstract moment of the not-being of Quantum ; resolves itself in itself; and thus it relates itself to its beyond as to its infinity, according to the qualitative moment of the opposition. Secondly, however, Quantum is continuous with this beyond : it is the very being of Quantum to be its own Other, to be self-external ; and thus this external part is equally nothing else than Quantum: the beyond, or the infinite, itself is a Qjiantum. Thus the beyond has been recalled from its flight, and the infinite has been reached. But now this “beyond” which has become “hither” once more is a Quantum; and, therefore, only a new limit has now been posited. This is a Quantum, and itself has again fled from itself; being a Quantum, it is beyond itself, and has repelled itself into its own not-being and its own beyond, which in turn becomes Quantum in the same perpetual manner in which Quantum repels itself from itself into beyond.
The continuousness of Quantum with its Other gives rise to the combination of the two in the expression of the infinitely great and the infinitely small. Both still contain the determina¬ tion of Quantum, and, therefore, are variable ; and the absolute determinateness, which would be a Being-for-Self, is thus not reached. This self-externality of the determination is posited in the double infinite, whose terms are in the opposition of “more” and “less,” namely, the infinitely great and the in¬ finitely small. In each. Quantum is preserved in perpetual opposition to its beyond. However much the great is extended, it shrinks into insignificance; it refers to the infinite as to its non-being, and thus the opposition is qualitative ; the extended Quantum has wrested nothing from the infinite, which remains its not-being as before. Or again, Quantum by being enlarged does not approach the infinite, for the distinction between Quantum and its infinity has this for essential moment, that it is not a quantitative distinction. This is only the expression of the contradiction brought under narrower consideration: the infinitely great is defined as great (that is, Quantum) and infinite (that is, not Quantum). — Similarly, the infinitely small, as small, is Quantum, and, therefore, remains qualitatively (that is, absolutely) too great for the infinite, and is opposed to it. In both is preserved the contradiction of the Infinite Progress, which was supposed to have found its goal in them.
This infinity which, as the beyond of the finite, is determined as stable, may be described as the “bad quantitative infinity.” Like qualitative bad infinity, it is an infinite alternation between the two members of a persistent contradiction, from the limit to its not-being, and from there back once more to the limit. In the quantitative progress the term towards which the progress moves is not indeed abstractly and altogether Other, but is merely a Quantum posited as different ; but it remains in opposition to its negation in a similar manner. The progress, too, therefore, is no forward advance but a repetition of the same cycle, of positing and transcending still repeated; an impotence of the negative which by the very act of transcending is ever again faced with that which it had transcended. Two members are connected in such a way that they must fly apart ; and in so doing they cannot part company, but even in flying from one another remain connected.
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Bad infinity, more especially in its form of the progress to infinity of the quantitative, is often considered sublime, and this continual surpassing of the limit (which really means an incapacity to transcend it), together with the perpetual relapse into it, has been held a kind of divine service. Philosophy has considered it as ultimate. And this progress has often furnished material for outbursts which have been admired as lofty achievements. Yet this modern sublimity does not exalt the object, for that escapes, but only the subject, which consumes such vast quantities. The meagre nature of this exaltation, which, still subjective, climbs up the ladder of the quantitative, is made clear when it is admitted that it vainly labours to reach an unattainable goal, which indeed to be gained must be quite otherwise attacked.
In the following outbursts of this nature it also appears whither this exaltation moves and where it ceases. Thus Kant cites it as sublime ( Critique of Practical Reason: end)
“when the subject rises in thought above the place which it occupies in the world of sense, magnifying into the infinitely great its relation, — a relation with stars beyond stars, worlds beyond worlds, systems beyond systems, where periodic move¬ ment, beginning, and duration are infinite in time as they are
in space. — Imagination fails before this progress into the immeasurably far, where the furthest world has one yet further, and past and future, however distant, have a more distant past and a more distant future beyond; thought fails before this image of the immeasurable; as a dream, where one goes down a long passage ever further and interminably further, without an end in sight, ends in a fall or in dizziness.”
This description condenses into rich imagery the content of quantitative exaltation : it also deserves praise for the truth¬ fulness with which it shows in what this exaltation ends: thought fails, the end is a fall and dizziness. And what causes thought to fail and produces the fall and the dizziness, is nothing else than the weariness of repetition, where the limit is made to vanish, return, and vanish again perpetually, and the hither to arise and perish in the beyond, and the beyond in the hither, one after the other and the one in the other; which causes a feeling of the impotence of this infinite and this ought , which would master the finite and cannot.
Haller’s description of eternity, too (called “the horrible description” by Kant), is often admired, though not for that aspect which constitutes its real merit: —
When from the horrible height Dizzy I look again on you.
When this heaping and piling of numbers and worlds is considered valuable as a description of eternity, it is over¬ looked that the poet himself declares this so-called horrible process vain and void, concluding that the true infinite reveals its presence only after this empty infinite progress has been abandoned.
There have been astronomers who took pleasure in priding themselves upon the sublime nature of their science because it dealt with an immeasurable multitude of stars and immeasur¬ able extents of space and time, where distances and periods, already great in themselves, serve as units which being taken
however many times over themselves shrink into insignificance. The shallow amazement to which they here abandon them¬ selves, and those silly hopes of travelling from one star to another in a second life, and of thus acquiring into infinity similar new knowledge, were proclaimed by them one of the chief accounts of excellence in their science, — which in truth is admirable, not because of this quantitative infinity, but rather because of those relations of measure and those laws which reason recognizes in these objects; for they are the infinity of reason and those others the infinity of unreason.
question 178-b
Kant opposes another infinity to that which relates itself to external sensuous intuition, when
“the individual returns upon his invisible ego, and opposes the absolute freedom of his will as pure ego to all the terrors of fate and of tyranny, and, beginning with his nearest sur¬ roundings, allows them to vanish before him, next causes what seem most durable, worlds upon worlds, to collapse in ruins, and, alone, knows himself equal to himself.”
The ego in this loneliness has indeed achieved the Beyond, it has gained itself, is gathered upon itself, and is on the hither side : in pure self-consciousness absolute negativity, which ever flies in that progress above sensuous Quantum, is affirmed and made present. But while this pure ego fixes itself in its abstrac¬ tion and lack of content, it has opposed to and beyond itself existence in general, or the totality of the natural and spiritual universe. The same contradiction appears on which the infinite progress is based : namely, a term has returned upon itself and also, immediately, is self-external, is related to its Other as to its not-being; and this relation remains a longing, for the ego has, on the one hand, fixed before itself, its own hollow and unstable void, and, on the other, the full content of its beyond which remains present even in the negation.
Kant subjoins to these two examples of the sublime the remark “that astonishment (for the first and external kind) and reverence (for the second and inward) stimulate investi¬ gation but cannot replace the lack of it.” — He thus declares these sublimities to be unsatisfactory to reason, which cannot remain with them and the feelings connected with them, and cannot allow the name of ultimate to the beyond and the void.
The infinite progress has been considered ultimate chiefly in its application to morality. The opposition just quoted between finite and infinite as standing respectively for the manifold world and the ego exalted in its freedom, is at this point qualitative. The ego in determining itself attempts simul¬ taneously to determine Nature and to free itself from it: it thus relates itself to its Other through itself: and the Other, being an external existence, is a manifold and quantitative. The relation to what is quantitative itself becomes quantitative; and the negative relation of the ego to it, the power which the ego exercises over the non-ego (the world of the senses, and external Nature), is therefore imagined as meaning that morality can and ought to grow, and the power of sensuousness to wane. But the perfect adequacy of the will to the moral law is relegated to the progress into the infinite, that is, it is imagined as an absolute and unattainable Beyond; and this its unattainable nature is supposed to be the true sheet-anchor and proper consolation; for morality should be a fight; and it is such only where will is not adequate to the law, and the law therefore is its absolute Beyond.
In this opposition ego and non-ego or pure will and moral law, and Nature and the sensuousness of will are presupposed as entirely independent and indifferent to one another. Pure will has its peculiar law, which is in essential relation with sensuousness; and sensuousness and Nature have their own laws, which neither come from nor are adapted to will, nor yet, although different from it, have any essential relation to it : they are determined in themselves, finished and complete. But also both are moments of one and the same simple essence, namely, the ego : will is determined as standing in the relation of negative to Nature, so that it exists only in so far as this different sphere exists, which it transcends, but is itself touched and affected thereby in so doing. Nature, taken as the sensuous sphere of man, is an independent system of laws : it is therefore indifferent to limitation by any Other; even while it is being limited it preserves itself, enters independently into relation, and limits the will of law as much as the will of law limits it. — When will determines itself and transcends the otherness of a nature, and again, when this otherness is posited as existing, as continued into a state of becoming-transcended, and yet
as not transcended — these two are one act. This contradiction is not resolved in the infinite progress, but is on the contrary represented and affirmed as unresolved and incapable of resolution: the struggle between morality and sensuousness is imagined as the self-existent, absolute relation.
Where it is thus impossible to master the qualitative oppo¬ sition between finite and infinite and to seize the idea of true will, or substantial freedom, recourse is had to magnitude as mediation, for magnitude is the qualitative element transcended, or distinction become indifferent. But both the members in opposition remain, qualitatively distinct, at bottom, so that when they are mutually related as quanta each immediately is made indifferent to this change. Nature is determined by ego and sensuousness by the will to the good, and the change thus produced therein is only a quantitative difference and one which allows it to remain as it is.
In the more abstract exposition of the Kantian philosophy, or at least of its principles, which is found in Fichte’s Doctrine of Science, the infinite progress similarly is the basis and the ultimate. The first principle of the exegesis, ego = ego, is fol¬ lowed by another which is independent of it, namely, the opposition of the non-ego; and the relation of these two is immediately taken as being a quantitative difference, non-ego being partly determined by ego and partly not. Thus the non¬ ego prolongs itself into its not-being, in such a manner that it remains opposed to this latter in it as being not yet tran¬ scended. Thus after the contradictions herein contained have been developed in the system, the final result is the same relation as was the beginning ; the non-ego remains an infinite “resistance-principle” ( Anstoss ), an absolute Other; the ulti¬ mate relation between it and the ego is the infinite progress, longing and yearning, — the same contradiction with which a beginning was made.
The quantitative is determinateness posited as transcended : for this reason it was thought that much or rather everything had been gained for the Unity of the Absolute, for Mono¬ substantiality, when contrariety in general was reduced to a merely quantitative difference. For a time it was a main thesis of modern philosophy, that all contrariety is only quantitative ; opposite determinations are of the same essence and content,
a crewmate finds a vast ocean of as-yet undiscovered alien lifeforms outside. one of them attacks him and sits on his face. he returns to the ship seeking medical aid. he will die if left outside the ship.
they are real sides of the opposition, in so far as each contains both the latter’s determinations and both its factors ; only, one factor, matter or energy, preponderates on one side, one on the other, or one is present in greater quantity or more intense degree on the one side than on the other. In so far as different kinds of matter and energy are assumed, the quantitative difference rather confirms and completes their externality and indifference both in relation to one another and to their unity. The distinction of absolute unity is supposed to be merely quantitative ; the quantitative is transcended immediate deter¬ minateness, but still, as negation, it is incomplete and first negation only, not infinite nor negation of negation. — Being and Thought are imagined as quantitative determinations of absolute Substance, and therefore, being quanta, they become quite external and unrelated to one another; as, in a subor¬ dinate sphere, do carbon, nitrogen, and so on. It is a third factor, an external reflection, which abstracts from the differ¬ ence between them and recognizes their unity, which is inner and only in itself, not for itself. Thus in fact this unity is imagined merely as of the first and immediate kind, or as Being which in its quantitative difference remains equal to itself, but does not through itself posit itself as equal to itself: it is thus not conceived as negation of negation, or infinite unity. It is only in the qualitative opposition that posited infinity or Being-for-Self appears ; and quantitative determination passes over — as will be seen immediately — into the qualitative.
It was mentioned above that Kant’s antinomies are repre¬ sentations of the opposition between finite and infinite in a more concrete shape and applied to more specialized substrata of imagination. The antinomy there considered contained the opposition of qualitative finitude and infinity. In another (the first of the four cosmological antinomies) it is rather the quantitative limit of which the self-conflicting nature is con¬ sidered. I shall therefore proceed here to the examination of this antinomy.
It deals with the spatial and temporal limitation or un- limitedness of the world. — Equally well the contradiction might
have been considered with reference to time and space them* selves, for the antinomy of limitation and unlimitedness remains in them unchanged, whether they be relations of things or only forms of intuition.
The closer analysis of this antinomy will also show that the two propositions with their proofs (which, like those considered above, are apagogic) come to nothing except these two simple and contrary assertions : “there is a limit,” and “the limit must be overpassed.”
“The world has a beginning in time, and spatially, too, is limited.”
One part of the proof, which deals with time, assumes the opposite : —
“If the world has no beginning in time, then an eternity has passed up to any given point of time; hence an infinite series of successive world-states has passed. But the infinite nature of a series consists precisely in this, that it can never be completed by successive synthesis. Thus an infinite world- series which also has elapsed, is impossible, and, therefore, a beginning of the world is an essential condition of its existence,— which was to be proved.”
The remaining part of the proof, which deals with space, is based upon time. It would require an infinite time to com¬ prehend the parts of a spatially infinite world, and this time would have to be regarded as expired in so far as the world in space is to be regarded not as a process but as a given whole. But it was shown of time in the first part of the prool that it is impossible to assume the expiration of an infinite period of time.
But it is immediately evident that it was unnecessary to make the proof apagogic, or, indeed, to give any proof, since it contains immediately for basis an assertion of that which was to be proved. For each, or any, point of time is here assumed, up to which an eternity has expired ( — eternity here meaning merely infinite time in the bad sense). Now a given point of time means nothing else than a definite limit in time. In the proof, therefore, a limit of time is assumed to exist; which is precisely what was to be proved. For the thesis is, that the world has a beginning in time.
what do you do?
There is only this one distinction, that the limit of time which is assumed is “now” as end of an already expired period, while that which is to be proved is “now” as beginning of the future. But this distinction is immaterial. “Now” is taken as the point where an infinite series of consecutive world-states has ended : it is taken as end, or qualitative limit. If this “now” were considered merely as quantitative limit, which remains fluid and not only must be overpassed but itself is nothing but an overpassing of itself, then the infinite time-series in it would not be expired but would continue to flow; and the argument of the proof would vanish. However, the point of time is taken as qualitative limit for the past, but this is also the beginning of a future (for in itself every point of time is the relation of past and future), and, further, is its absolute or abstract beginning, that is, that which was to be proved. It is irrelevant that this future and this its beginning are pre¬ ceded by a past; in so far as this point of time is qualitative limit — and this is implied in the determination of expired or completed, that is, not self-continuing — time in it is broken off short, and that past is without relation to this time which could be called future only with reference to that past, and consequently when unrelated becomes time in general, which has an absolute beginning. But if— as is the case — through the “now” (or given point of time) it were linked to the past, and so determined as future, then this point of time too, seen from the other side, would be no limit ; the infinite time-series would continue itself in that which was before called future and would not be, as was assumed, completed.
In truth time is pure Quantity : the point of time at which it is supposed to be interrupted, which is employed in the proof, is simply the self-transcending Being-for-Self of the Now. The proof effects nothing, except to present to imagination the absolute limit of time, which is asserted in the Thesis, as a given point of time, and to take it boldly as a complete or abstract point; — a popular determination which sensuous imagination easily allows to pass as limit, and thus allows the proof to assume what before was set forth as to be proved.
“The world has no beginning and no limits in space, but is infinite both with regard to time and to space.”
“Let the world be assumed to have a beginning. Now the beginning is an existence preceded by a time in which the thing does not exist; therefore a time must have preceded in which the world did not exist; that is, an empty time. Now in an empty time nothing can arise, for no part of such a time has in itself, and in preference to any other, the differentiating condition of its existence or non-existence. Therefore any series of things can begin in the world, but the world itself cannot begin and with regard to past time is infinite.”
Like the others this apagogic proof contains the direct and unproved assertion of that which it ought to prove. At first it assumes a Beyond of the existence of the world, namely, empty time; but next it also continues the existence of the world beyond itself into this empty time, thus cancelling it, and continuing this existence into infinity. The world is an existence; the proof presupposes that this existence arises, and that the arising has a temporally precedent condition. But the Antithesis just consists in this, that there is no unconditioned existence and no absolute limit, but that the existence of the world always demands a precedent condition. — What was to be proved is thus already present in the proof as an assump¬ tion. — Further, the condition is now looked for in empty time, which means that it is taken as temporal and therefore as an existence and as limited. Thus altogether the assumption is made that the world as existence presupposes another con¬ ditioned existence in time, and so to infinity.
The proof with regard to the spatial infinity of the world is the same. The spatial finitude of the world is posited in the apagogic manner : “the world would thus be in an empty and unlimited space and would stand in a relation to it; but such a relation of the world to no object is Nothing.”
There again the proof directly presupposes what was to be proved. It is directly assumed that the limited spatial world must be in an empty space and stand in a relation to it, that is, that it must be overpassed; — a process leading on the one hand into the void, the beyond and not-being of the world, and, on the other, meaning that it stands in a relation to the void, that is, that it continues itself into it and therefore that the beyond is to be imagined as filled with world-existence.
The spatial infinity of the world which is asserted in the Antithesis is just ( a ) empty space and ( b ) the relation of the world to it, that is, the continuity of the world in it, or the filling of empty space; and this contradiction — that space is empty and also is full — is the infinite progress of Existence in space. And this contradiction itself, the relation of the world to empty space, is directly made the basis of the proof.
The Thesis and the Antithesis and their proofs thus represent nothing but the contrary assertions that there is a limit, and that this limit equally is cancelled, that the limit has a beyond, but that it is in relation with the latter and must pass over to it; which again gives rise to such a limit which is no limit.
The solution of these antinomies, as with the first, is tran¬ scendental, that is, it consists in the assertion of the ideality of space and time as forms of intuition, — in this sense, that the world in itself is not in contradiction with itself and does not transcend itself, but that consciousness in its intuition and in the relation of intuition to understanding and reason is a self-contradictory essence. But it is an excessive tenderness towards the world to remove from it the contradiction and to transfer it into Spirit or Reason, allowing it to remain there unresolved. In point of fact it is Spirit which is strong enough to support the contradiction, but it is also Spirit which knows how to resolve it. As for the so-called world (whether it be called objective or real world, or, according to the transcendental idealism, subjective intuition and sensuousness determined by the category of understanding), it never and nowhere is without the contradiction, but, since it cannot support it, is subject to arising and passing away.
- let him on board and bring him to the medbay
- breathe in his spores and invite him to dinner
- blow up the ship immediately
- keep him outside and go home
- find more aliens and bring them aboard the ship
- go back to bed as you are not paid for this
- kill him without remorse
answer:
1. Infinite Quantum, as infinitely great or infinitely small, is, in itself, the infinite progress ; being either great or small, it is Quantum ; and it is also the not-being of Quantum. Hence the infinitely great and the infinitely small are pictures of the imagination, which, closely considered, prove mere mist and shadow. But in the infinite progress this contradiction is present explicitly, and with it that which is the nature of Quantum which has reached its reality as intensive magnitude and is
now posited in its determinate existence as it is in its concept. This identity must now be considered.
Quantum as degree is simple, self-referred, and self-deter¬ minate. Through this simplicity, otherness and determinateness are cancelled in it, and therefore the latter is external to it: it has its determinateness outside it. This self-externality is, first, the abstract not-being of Quantum in general, or bad infinity. But, further, this not-being is itself a magnitude; Quantum continues itself into its not-being, for it is in its externality that it has its determinateness; therefore this ex¬ ternality, too, is itself Quantum ; and its not-being, or infinity, is thus limited, that is, this beyond is transcended and is determined as Quantum, which thus, in its negation, is gathered to itself.
But this is just what Quantum as such is in itself. For it is itself because it is outside itself: externality constitutes that by virtue of which it is Quantum and is with itself. Thus in the infinite progress the concept of Quantum is posited.
If we take it first in its abstract determinations as we find them, it contains the transcendence of Quantum, but also of its Beyond ; there is thus the negation of Quantum as well as the negation of this negation. In its truth it is the unity in which these are contained, but as moments only. — This truth is the solution of the contradiction of which the progress is the expression, and its nearest meaning is therefore the rehabili¬ tation of that concept of magnitude where it is indifferent or external limit. In the infinite progress as such the only reflection made generally is that every Quantum, however great or small, can vanish, that progress may take place beyond it; but the reflection, that this its transcendence, the Beyond, and the bad infinite also vanish, is generally not made.
The first transcendence, the negation of Quality in general, whereby Quantum is posited, is already in itself the tran¬ scendence of negation, Quantum being transcended qualitative limit and therefore transcended negation; but also it is this in itself only; it is posited as a Determinate Being, and hence its negation is established as the infinite, as the Beyond of Quantum, which takes up the place of Hither or of immediate : thus the infinite is determined only as first negation, and as such it appears in the infinite progress. But it has been shown
that this contains more, namely, the negation of negation or that which is in truth the infinite. This was looked at before in such a way that the concept of Quantum was thus restored ; and this restoration means, first, that its Determinate Existence has received a closer determination : for Quantum determined according to its concept has now arisen, and this differs from immediate Quantum ; externality is now its own opposite ; it is posited as a moment of magnitude itself, and Quantum is posited as having its determinateness in another Quantum through the mediation of its not-being, namely, of infinity; that is, it is qualitatively that which it is. However, this com¬ parison of the concept of Quantum with its existence belongs rather to our reflection, to a relation which is not yet here at hand. The nearest determination is this, that Quantum having returned to Quality is now determined qualitatively. For Quality, which is its characteristic, is externality or indifference of determinateness ; and it is now posited as, in its externality, being itself, and as self-relating, having its determination in simple self-unity, that is, qualitatively. — This Qualitative is also more closely determined, it is determined as Being-for-Self; for the relation to itself which it has reached has been evolved from mediation, from the negation of negation. Quantum is infinite and is determined for itself; and this is no longer external to it, but within itself.
The infinite, which in the infinite progress had the mere empty meaning of a not-being, a Beyond not reached but searched for, is, in fact, just Quality. Quantum as indifferent limit passes beyond itself into the infinite, and in so doing seeks after nothing else than determinateness-for-self, the qualitative moment which hitherto, however, is no more ' than Ought. Quantum is Quantum, because of its indifference to the limit and because, therefore, it has no self-subsistent determinate¬ ness and so overpasses itself : this its overpassing is to be negated and its absolute determinateness is to be found in the infinite.
And, quite generally, Quantum is Quality transcended ; but Quantum is infinite, overpasses itself, and is its own negation : thus this overpassing is in itself the negation of negated Quality, and its restitution; and so it is posited that externality, which appeared as a Beyond, is determined as own moment to Quantum.
Quantum thus is posited as self-repelled : there thus are two Quanta, which, however, are transcended and exist only as moments of one unity: this unity is the determinateness of Quantum. — Quantum then, related to itself as indifferent limit in its externality and therefore qualitatively posited, is the Quantitative Ratio. — In Ratio, Quantum is external to and different from itself: this its externality is the relation of one Quantum to another, each of which has value only in this its relation to its Other; and this relation constitutes the deter¬ minateness of Quantum, which exists as such a unity. Here its determination is not indifferent, but qualitative: in this its externality it has returned upon itself, and in it is that which it is.
The mathematical infinite is interesting, first, because its introduction has widened the scope of mathematics and has led to important results therein ; next it is remarkable because this science has not yet succeeded in vindicating this use of it conceptually (“concept” being here taken in its proper mean¬ ing). Ultimately, such vindications are based on the correct¬ ness ( demonstrated on other grounds) of the results achieved by means of this determination, and not on the clarity of the object and of the process which produces the results: indeed, the process itself is admitted to be faulty.
This in itself is a flaw : such a procedure is unscientific. But it also involves the drawback that mathematics, not knowing the nature of this its instrument (and this, because it cannot cope with the metaphysics and critique of it), could not deter mine the sphere in which it must be applied, and could not safeguard itself against misapplication.
In its philosophical aspect, however, he tmathematical infinite is important because the concept of the true infinite is, in fact, its basis : it stands on a much higher level than the so-called metaphysical infinite, whence objections against the former are launched. The science of mathematics often has no refuge from these objections except in denying the com¬ petence of metaphysics, asserting that it has nothing to do with
this science and need not trouble about its concepts if only it acts consistently within its own sphere. It need not consider truth in itself, but only what is true in its own domain. And metaphysics, thoiugh it opposes them, cannot deny nor in¬ validate the brilliant results of the employment of the mathe¬ matical infinite; while mathematics cannot achieve clearness about the metaphysics of its own concept, nor, therefore, about the derivation of the methods which the employment of the infinite necessitates.
If it were this single difficulty of the concept in general which oppressed mathematics, it could without ceremony neglect this, at least in so far as the concept is more than an expression of the essential determinations of a thing, that is, those which determine it for the mere Understanding : and mathematics has not allowed such determinateness to be lacking in precision ; for it is not a science which is concerned with the concepts of its objects or has to create its content by the development (even if this were only through a process of argumentation) of the concept. But in the method of the mathematical infinite mathematics finds a radical contradiction to that very method which is characteristic of itself, and on which it rests as a science. For the calculation of the infinite admits of, and demands, modes of procedure which mathematics, when it operates with finite magnitudes, must altogether reject, and at the same time it treats these infinite magnitudes as finite Quanta, seeking to apply to the former those same methods which are valid for the latter. The most important step in the development of this science is to have imposed on transcendent determinations and the treatment of these the form of ordinary calculation.
In spite of this conflict in its operations, mathematics shows results which agree exactly with those found by the true mathematical method, the geometrical and analytic method. However, this does not apply to all results; and the end for which the infinite is introduced is not to shorten the beaten road, but to reach results to which it cannot lead. And also the success does not justify the manner of road. Now, this manner of calculating the infinite is plainly oppressed by the apparent inexactitude in which it involves itself when, first, it increases finite magnitudes by an infinitely small amount, and then partly retains these during the further processes, but voff. 1 1
partly also neglects. This procedure has this strange quality, that in spite of the admitted inexactitude a result is achieved which is exact, — not merely tolerably or to such a degree that the difference might be neglected, but absolutely. But in the process itself which precedes the result we cannot do without the suggestion that a certain quantity is not equal to nil but is so inconsiderable that it may be neglected. But where mathematical exactitude is meant, the distinction between greater and less accuracy quite disappears, just as in philosophy there can be no question of greater or less probability, but of truth alone. If the method and employment of the infinite are justified by success, it is not as superfluous to demand notwithstanding this a justification as to ask in the case of the nose for a demonstration of the right to use it. For mathematical cognition is a scientific cognition, and therefore the proof is essential ; and as regards the results, too, it is the fact that a strictly mathematical method does not stamp all with the mark of success, — which mark in any case is only external.
It is worth while more closely to consider the mathematical concept of the infinite, together with the more remarkable among the attempts designed to justify its employment and to remove the difficulty by which the method feels itself oppressed. The consideration of these justifications and determinations of the mathematical infinite, which I shall undertake at some length in this Observation, will also throw the clearest light on the nature of the true concept itself, and will show how a confused notion of this latter was present as a foundation for those processes.
The mathematical infinite is commonly determined as a magnitude than which there is no greater (when it is deter¬ mined as the infinitely great) or no smaller (when it is deter¬ mined as the infinitely small) ; or as one which is greater (in the former) or smaller (in the latter instance) than any given magnitude. — It is true that in this definition the true concept is not expressed, but only, as has already been remarked, -the same contradiction as is contained in the infinite progress; however, let us see what it contains in itself. A magnitude is defined in mathematics as something which can be increased and decreased : that is, then, as an indifferent limit. Now since the infinitely great or small is such as cannot be
increased nor decreased, it is in fact no longer a Quantum as such.
-grombo
This consequence follows necessarily and immediately. But it is just the reflection that Quantum has been transcended which is generally not made (and in this Observation I designate as “finite Quantum,” Quantum in general, as we find it): and it is just this which constitutes the difficulty for ordinary thought; for it is demanded that Quantum, in so far as it is infinite, be thought of as transcended and as something which is not a Quantum, although its quantitative determinateness remains.
We may here quote Kant’s judgment about this determina¬ tion :J he finds that it does not agree with the common notion of an “infinite whole.” “According to the ordinary concept, that magnitude is infinite beyond which there can be no greater magnitude (that is, beyond the multitude of a given unit which it contains). But no multitude is the greatest, because it is always possible to add one or more units. — But the notion of an infinite whole does not imply any magnitude, and there¬ fore the concept of an infinite whole is not the concept of a maximum (or minimum) : this concept rather expresses the relation of the whole to any given unit, with respect to which it is greater than any number. The infinite would be greater or less according as this unit was taken to be greater or smaller; but infinity, which consists only in the relation to this given unit, will always remain the same, although of course the absolute magnitude of the whole cannot be known through it.”
Kant disapproves when infinite wholes are regarded as a maximum, as the entire multitude of a given unit. Maximum or minimum as such still appears as a Quantum, or a multitude. And this idea cannot avert the conclusion, mentioned by Kant, which leads to a greater or lesser infinite. And generally, so long as the infinite is imagined as Quantum, the distinction between greater and less applies to it. But this criticism has no force against the concept of the true mathematical infinite, the infinite difference, for this no longer is a finite Quantum.
1 Observation to the Thesis of the First Cosmological Antinomy, in the Critique of Pure Reason.
Opposed to this is Kant’s concept of infinity, which he calls the true and transcendental, — namely, “that the successive synthesis of the unit, in measuring a Quantum, can never be completed.” A Quantum in general is supposed to be given : it is demanded that, by synthetizing the unit, this be made into an amount, into a Quantum which can definitely be indicated; and it is asserted that this synthetizing can never be completed. Clearly, it is only the progress to infinity which has here been posited; only it is figured in a transcendental, that is really in a subjective and psychological, manner. It is demanded that in itself Quantum be complete ; but (it is said) in the transcendental manner, that is, in the subject, which gives it a relation to a unit, the only determination of the Quantum which arises is such as is incomplete and altogether infected with a Beyond. The contradiction which magnitude contains still remains the limit of our progress; but it is distributed between object and subject, limitedness being attri¬ buted to the former, and to the latter the passing beyond every determinate measure which it may have reached at any stage into the bad infinite.
On the other hand it was stated above that the determina¬ tion of the mathematical infinite, as it is used in the higher analysis, corresponds to the concept of the true infinite: the comparison of the two determinations will now be undertaken in its more detailed development. — And, first, as regards true infinite Quantum, it determined itself as intrinsically infinite; and it is so because, as was seen, finite Quantum (or Quantum in general) and its Beyond, the bad infinite, are alike trans¬ cended. Quantum transcended has thus returned into sim¬ plicity and self-relation, and this not merely as did Extensive Quantum when it passed over into Intensive Quantum; for this latter has its determinateness only in itself in an external multiplicity, towards which it is required however to be indifferent and from which it must be distinct. But Infinite Quantum contains in itself, first externality, and then the negation of this; and thus it no longer is a finite Qpantum or determinateness of magnitude having a determinate exist¬ ence as Quantum, but is simple, and therefore exists only as moment : it is determinateness of magnitude in a qualitative form, and its infinite nature consists in the fact that it is
qualitative determinateness. — As moment, then, it is essentially at one with its Other only as determined through this its Other, that is, it has significance only with reference to some¬ thing which stands in a relation towards it. Apart from this relation it is nil,— just because the Quantum as such has to be indifferent towards the relation, whilst in the relation it is an immediate constant value. In the relation, as a mere moment, it is not indifferent for itself; since it is at the same time a quantitative determinateness, it exists in infinity (regarded as Being-for-Self) only as a “For-One.”
It will appear that the concept of the infinite as here abstractly expounded is the basis of the mathematical infinite : the concept itself will become clearer if we consider the several steps of the expression of Quantum, as a moment of relation, beginning with the lowest step where it is still also Quantum as such, and proceeding to the higher where it receives the meaning and expression of real infinite magnitude.
Let us first take Quantum in a ratio, such as is a fraction.
Such a fraction, for instance -, is not a Quantum like 1, 2, 3
i’d forgotten there’s a bit of an analog horror dimension to it, too! i really feel like it’s got everything
and so on; it is an ordinary finite number, but not absolute, like integers: as fraction it is determined mediately by two other numbers which stand in the relation of Amount and Unit, the Unit itself being a determinate Amount. However, if we abstract from this closer mutual determination of these, and consider them only according to their fate as Quanta, in the qualitative relation in which they here are, 2 and 7 are Quanta otherwise indifferent: here, however, they act only as moments, one of the other, and, therefore, of a third (namely, that Quantum which is called the Exponent) : here it is not their status as 2 and 7, but their mutual determinateness, which matters. In their place 4 and 14, 6 and 21, and so on to infinity, might equally well be put. They thus herewith assume a qualitative character. If they counted as mere Quanta, then 2 and 7 are, the one just 2 and the other just 7 ; and 4, 14, 6, 21 and so on are simply other than those numbers, and, in so far as they would be absolute Quanta, they could not be sub¬ stituted for one another. But, since 2 and 7 are not counted according to their determinateness as such Quanta, their indifferent limit is transcended ; and thus in this aspect they
contain the moment of infinity ; for not only is each number not itself, but also their quantitative determinateness remains ; it remains, however, as qualitative in itself, and depends on their value as related. An infinite number of others could be put in their place, and yet the value of the fraction, owing to the determinateness possessed by the relation, is not varied.
But infinity is still not adequately represented by a fraction, and this because the two sides of the fraction, 2 and 7, can be taken apart from the ratio, and are ordinary indifferent Quanta: the relation which subsists between them (namely, that of being in ratio, and of being moments) is external and indifferent to them. And their relation too is an ordinary Quantum, the exponent of the ratio.
The letters with which general arithmetic operates form the next universality into which numbers rise: they have not the peculiarity of bearing a determinate numerical value; they are only general symbols and indeterminate potentialities of any
determinate value. And therefore the fraction \ seems to be a
more fitting expression of the infinite, since a and b, taken apart from their relation to one another, remain indeterminate, and severally too have no special and peculiar value. — But, though these letters are posited as indeterminate magnitudes, their significance is that they are some finite Quantum. Thus they are the general image, but the image only of determinate number, and therefore with them too it is indifferent that they should be in a ratio, and they retain this value also when they are no longer in such ratio.
If we consider more closely what is present in Ratio, we find therein these two determinations, first that it is a Quantum, and secondly that this Quantum is not immediate, but contains the qualitative opposition; and also there remains in it that determinate and indifferent Quantum, because it returns upon itself from its otherness (the opposition), and there¬ fore is infinite itself. These two determinations are seen clearly developed in their distinctness from each other the following well-known form.
The fraction - may be expressed as 0-285714 1 + a -f a2 -f a3 + .
the fraction itself is called the sum or finite expression of it. If these two expressions be compared, then one, the infinite series, no longer represents it as a ratio, but in its aspect as Quantum (since it is a multitude of added units) or as Amount. — It is here irrelevant that the magnitudes of which it is supposed, as Amount, to consist are in turn themselves made up of decimal fractions, that is, of ratios ; for this fact applies only to the special kind of unit of which these magnitudes are composed, and not to the magnitudes as constituting an Amount; and, similarly, a multi-figured integer of the decimal system is counted essentially as Amount, and the fact is neg¬ lected that it consists of the products of a number and of the number ten and the powers of ten. And, similarly, it is here irrelevant that there are other fractions besides the one taken
of -, which, if converted into decimals, do not yield an infinite
-rachel
series; every fraction can be so expressed in a numerical system having a different unit.
Thus, in the infinite series which is to represent the fraction as Amount, its aspect of ratio vanishes; and simultaneously there vanishes that aspect which — as was shown above — gave to it infinity. Infinity, however, has now been introduced in another manner ; for the series itself is infinite.
Now it is self-evident what is the nature of the infinity of the series : it is the bad infinity of the progression. The series contains and illustrates the contradiction which would repre¬ sent as a mere Quantum or Amount, or as non-relational, that which is a ratio and includes a qualitative nature. The conse¬ quence of this is that the Amount expressed in the series is ever incomplete, so that what is posited must still be sur¬ passed if the determinateness demanded would be reached. The law of progress is known, and is implicit in the deter¬ mination of Quantum which is contained in the fraction, and in the nature of the form in which this determination has to be expressed. The Amount can indeed be made as exact as is necessary by a continuation of the series ; but the represen¬ tation through the series still remains an Ought; it is infected with a Beyond which cannot be transcended, because to express as Amount that which is based on qualitative determinateness, is an abiding contradiction.
This inexactitude, of which there is only an appearance in the true mathematical infinite, is actually present in this infinite series. These two kinds of mathematical infinite are as little to be confounded as the two kinds of philosophical infinite. In representing the true mathematical infinite the form of a series was originally used, and has again lately been invoked. But this form is not necessary for it; on the contrary the infinite of the infinite series differs from it essentially, as the sequel will show. Indeed, it is inferior even to the fraction as a form of expression.
For the infinite series contains bad infinity, since that which the series is designed to express remains as Ought; and what it does express is infected with a Beyond that never vanishes, and is distinct from what it is desired to express. It is infinite, not because of the number of its terms, but because they are incomplete, because the Other, which essentially belongs to them, is beyond them; that which it really contains (let the terms posited be as many as they will) is but something finite, in the proper sense, posited as finite — that is as something which is not what it ought to be. That, on the contrary, which is called the finite expression or sum of such a series, has nothing lacking : it contains fully the value which the series only seeks after; the Beyond is recalled from its flight; what it is, and what it ought to be, are not severed, but the same.
The differentia of these two is, more precisely, this : in the infinite series the negative is outside its terms, which are present only in so far as they count as parts of the Amount. On the other hand, in the finite expression, which is a ratio, the negative is immanent as being the mutual determinateness of the sides of the ratio; and this, being a return upon self or self-relating unity as negation of negation (for both the sides of the ratio exist only as moments), contains the determina¬ tion of infinity. — Thus, in fact, what is ordinarily called sum
^like ^ or ■— l -^j is 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 : it is designed to set forth in the form of sum what is in itself a ratio, and terms of the series exist not as terms of a ratio but of an aggregate. And, further, it is, more properly, the finite expression ; for it is an incomplete
aggregate and essentially has something lacking. According to its actual content it is a determinate Quantum, but also a Quantum less than it should be; and that which is lacking is then also a determinate Quantum. This lacking part is in fact that which is called the infinite part of the series, in this merely formal meaning, that it is a something which is lacking, or a not-being; according to its content it is a finite Quantum. It is only that which the series contains, plus that which is lacking, which constitutes the fraction, — that determinate Quantum which it too ought to, but cannot, be. — In the infinite series, too, the word “infinite” generally enjoys the reputation of something lofty and venerable; this is a kind of superstition, the superstition of the understanding; we have seen that it reduces itself rather to the determination of a deficiency.
We may still remark that the existence of infinite series which cannot be summed is a circumstance altogether external and contingent with respect to the form of the series. These contain a higher kind of infinity than do the series which can be summed: they contain incommensurability, — that is to say, the quantitative ratio which they contain cannot be repre¬ sented as Quantum even in the shape of a fraction. However, the form of series as such, which is theirs, contains the same determination of bad infinity that exists in the series which admits of being summed.
The terminological inversion remarked above in connexion with fractions and their equivalent series is found also in so far as the mathematical infinite — not that just mentioned, but the true one — has been called the relative infinite, and the ordinary metaphysical infinite on the other hand (by which is meant the abstract and bad infinite), the absolute. But, in fact, this metaphysical infinite is merely relative, because the negation which it expresses remains in the opposition of a limit only in such a way that the limit persists outside it and is not transcended by it; whereas the mathematical infinite has veritably in itself transcended its finite limit, because its Beyond is incorporated with it.
It is in this sense, in which it has just been demonstrated that the so-called sum or finite expression of an infinite series is in fact to be regarded as its infinite expression, that Spinoza opposes the concept of true infinity to that of bad, and
elucidates it by means of examples. His concept gains the greatest clarity if I join his observations on this subject to my exposition.
He begins by defining the Infinite as the absolute affirmation of the existence of any one Nature, and the Finite on the contrary as determinateness, or negation. For the absolute affirmation of an existence must be taken as meaning its self¬ relation, the fact that its existence does not depend on that of an Other. The finite, on the other hand, is negation, that is, cessation as meaning relation to an Other which begins beyond it. It is true that the absolute affirmation of an existence does not exhaust the concept of infinity ; for this concept also implies that infinity is affirmation not as immediate, but only as restored through the intro-Reflection of the Other, or as negation of the negative. But with Spinoza, Substance and its absolute unity have the form of an inert, that is, of a not self-mediating, unity,-- of a rigidity wherein the concept of the negative unity of the self (Subjectivity) has not yet found a place.
Spinoza explains the true infinite by means of a mathematical example (Epist. XXIX), namely, a space contained between two unequal circles which arc not concentric, while one lies within the other without touching it. It seems that he thought much of this figure and of the concept for which he used it as example: he made it the motto of his Ethics. — “Mathe¬ maticians,” he says, “conclude that the inequalities which are possible in such a space are infinite, not from the infinite multitude of parts (for its magnitude is determinate and limited, and I can posit similar spaces both greater and smaller), but because the nature of the matter surpasses any determinate¬ ness.” — Evidently Spinoza rejects that idea of the infinite according to which it is imagined as multitude or as incomplete series, reminding us that here in the space of the example the infinite is not beyond, but is present and complete ; this space is limited, and is infinite, “because the nature of the matter surpasses all determinateness,” because, that is, the magni- tudinal determination which it contains cannot also be repre¬ sented as a Quantum, or (to use the expression of Kant quoted above) because the synthesis cannot be completed into a (discrete) Quantum. — How in general the opposition of con-
tinuous and discrete Quantum leads to the infinite, must be explained in a later Observation. — Spinoza calls this serial infinite “the infinite of imagination,” but the infinite as self¬ relation “the infinite of thought,” or infinitum actu. For it is actually (actu) infinite, because it is complete and present in itself. Thus the series 0-285714 ... or i+a + «* + a9 is merely the infinite of fancy or opinion, for it has no actuality
actual infinite, and not only is what the series is in those of its terms which are given, but has also that which the series lacks and is what the latter only ought to be. The fraction
- or — - — is also a finite magnitude, like Spinoza’s space 71 — a
enclosed between two circles, with its inequalities; and, like that space, it may be increased or decreased. But this does not result in the absurdity of an infinite which may be greater or less ; for this Quantum of the whole does not touch the ratio between its moments, the “nature of the matter,” that is, the qualitative determination of magnitude. That which the infinite series presents to us is also a finite Quantum; but, further, it is defective. — But fancy stands still at Quantum as such, and does not reflect upon the qualitative relation that is the basis of the incommensurability which faces it.
That incommensurability which is implied in Spinoza’s example includes in a general way the functions of curves, and, more exactly, leads to the infinite which mathematics has introduced with such functions and, generally, with the functions of variable magnitudes ; this is the true mathematical or qualitative infinite, which also Spinoza had in mind. We will now discuss this determination in greater detail.