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§ 538
The mathematical infinite has expanded the bounds of science even though mathematics has not justified the use of the infinite as a Notion.
the justifications are not based on the clarity of the subject matter and on the operation through which the results are obtained, for it is even admitted that the operation itself is incorrect.
§ 539
Unfortunately mathematics has not "mastered the metaphysics and critique of the infinite."
§ 540
But philosophy is interested in the mathematical infinite because philosophy shows that there is a genuine metaphysical infinite that contradicts the mathematical one, even though the mathematical one produces accurate mathematical results.
§ 541
mathematics, in the method of its infinite, finds a radical contradiction to that very method which is peculiar to itself and on which as a science it rests. For the infinitesimal calculus permits and requires modes of procedure which mathematics must wholly reject when operating with finite quantities, and at the same time it treats these infinite quantities as if they were finite and insists on applying to the former the same modes of operation which are valid for the latter.
In calculus, a quantity is taken both as finite and infinitesimal.
it is a cardinal feature in the development of this science that it has succeeded in applying to transcendental determinations and their treatment the form of ordinary calculation.
§ 542
The mathematical use of the infinite, even if metaphysically ungrounded, produces accurate results. However
1) it does not always produce accurate results, and also when the mathematical infinite is employed it is only because the ordinary method falls short, and
2) success does not by itself justify the mode of procedure. This procedure of the infinitesimal calculus shows itself burdened with a seeming inexactitude, namely, having increased finite magnitudes by an infinitely small quantity, this quantity is in the subsequent operation in part retained and in part ignored.
And yet despite the fact that the procedure of calculus is inexact, its results are always no less than perfectly precise. Nonetheless, we have every right to demand a justification for its use, namely, for a proof demonstrating its legitimacy.
§ 543
Hegel will now examine some attempts to justify the mathematical infinite.
§ 544
The usual definition of the mathematical infinite is that it is a magnitude than which there is no greater (when it is defined as the infinitely great) or no smaller (when it is defined as the infinitely small), or in the former case is greater than, in the latter case smaller than, any given magnitude.
Mathematics defines a magnitude as "that which can be increased or diminished." But, since the infinitely great or small can be neither increased nor diminished, it cannot be a quantum.
§ 545
In order to think of a quantum as infinite, then, we must consider it as "sublated;" which is to say that we must consider it as "something which is not a quantum but yet retains its quantitative character."
§ 546
Hegel quotes Kant's definition of the infinite whole:
According to the usual concept, a magnitude is infinite beyond which there can be no greater (i.e. greater than the amount contained therein of a given unit); but there can be no greatest amount because one or more units can always be added to it. But our concept of an infinite whole does not represent how great it is and it is not therefore the concept of a maximum (or a minimum); this concept rather expresses only the relation of the whole to an arbitrarily assumed unit, with respect to which the relation is greater than any number. According as this assumed unit is greater or smaller, the infinite would be greater or smaller. The infinity, however, since it consists solely in the relation to this given unit, would always remain the same, although of course the absolute magnitude of the whole would not be known through it.
We might note here that Kant excludes the notion of a maximum or minimum limit for the infinite, when Spinoza on the other hand proposes a sort of infinite set within such limits.
§ 547
Hegel comments that Kant's objection to infinite wholes being regarded as a maximum or minimum, that is, as a completed amount of a given unit, because we can always conceive of something greater or smaller than any given limited amount.
§ 548
Kant offers his own "transcendental" concept of the infinite:
'that the successive synthesis of the unit in the measurement of a quantum can never be completed.'
This is a synthesis of the unit into an amount that is never completed. But Hegel says that this is just a progressive infinite that the subject constructs psychologically:
in itself the quantum is supposed to be completed; but transcendentally, namely in the subject which gives it a relation to a unit, the quantum comes to be determined only as incomplete and as simply burdened with a beyond.
Thus Kant does not advance past the contradiction of infinity.
§ 549
Hegel will now compare the mathematical infinite with the genuine infinite.
§ 550
The infinite quantum contains within itself both externality as well as the negation of externality, hence it is no longer a finite quantum.
it is simple, and therefore only a moment. It is a quantitative determinateness in qualitative form; its infinity consists in its being a qualitative determinateness.
§ 551
This Notion of the infinite is the basis for the mathematical conception of it, which Hegel will explicate by discussing the quantum as a moment of a ratio.
§ 552
In fractions, quanta are in relation to each other, for example, in 2/7, the 2 and the 7 are both quanta on their own. But any quanta can be substituted into such a ratio relation, so the 2 and the 7 are "indifferent quanta" because it is arbitrary to the relation itself which quanta are used, so long as the ratio comes out the same: we might instead use 4 and 14, 6 and 21, and so on to infinity. Hence both the 2 and the 7 are merely moments of this sort of ratio relation. For this reason, the 2 and 7, although each alone are quantitative, when they come into a ratio relation which has an infinity of other moments, they take on a qualitative nature.
§ 553
But this ratio relation is an inadequate representation of infinity, because we can still isolate the 2 and the 7 and take them as ordinary finite quanta.
§ 554
Variables, represented by alphabetical letters, "are only general symbols and indeterminate possibilities of any specific value." So the fraction a/b seems to be a better expression of the infinite, because both a and b, taken alone, are undetermined. However, even so, these indeterminate magnitudes still are supposed to represent finite magnitudes, even if they are not specified..
§ 555
The ratio relation we have been considering contains quantitative components set into a qualitative opposition (the relationship of one quantity being against the other is qualitative).
§ 556
Hegel notes that
The 1/(1 - a) as 1 + a + a2 + a3 etc. no longer represents the ratio as a fraction but rather as a sum of an infinite series of terms, which now represents a "quantum as an aggregate of units added together, as an amount."
§ 557
We noted previously that fractions are infinities of a certain sort (§ 552), and we see that when we convert them to summation series, they become infinities of another sort.
§ 558
The infinite series is a "spurious infinity" of progression because it takes the qualitative relation of the terms set against each other in a ratio and represents them as though it were just a mere quantum or amount with a quantitative nature.
The consequence of this is that the amount which is expressed in the series always lacks something, so that in order to reach the required determinateness, we must always go further than the terms already posited.
That is to say, there is an irreducible (irrational) quality to the ratio that cannot be rendered into a determinate quantity, so when we do so, we have to progressively reiterate the representation process to infinity. But this is not true infinity in itself, it is rather just a process that is carried out to infinity.
§ 559
The fractional infinite (2/7) seems inexact but really it is not, because their irrational relation can be taken together in this form, but in the infinite series, inexactitude is built into the process of determining the value.
§ 560
The infinite series is only incomplete because each successive determination points beyond it, but what is really present in the series is an exact amount.
§ 561
In the infinite series, the negation that continually sets something beyond the series lies outside the terms. In the infinite ratio, the negative element that allows for an infinity of substitutions is a part of the terms, because when we negate the 2/7 to get 4/14 and then to get 6/21, each time we negate the negation from within; for, both sides of the ratio are merely moments. But the infinite series is a sum, which represents the ratio as an aggregate and is hence a finite expression.
§ 562
The infinite series is a higher kind of infinity than series which can be summed, because infinite series bear an incommensurability, or "the impossibility of representing the quantitative ratio contained in them as a quantum, even in the form of a fraction." However, the way this incommensurability is represented in an infinite series is as a sum and hence as a finite figure.
§ 563
The metaphysical infinite is relative because its limit is posited as outside it, whereas the mathematical infinite is absolute because its limit is sublated and united with it.
§ 564
Spinoza opposes the concept of true infinity to that of spurious infinity by illustrating with examples the way that the "so-called sum or finite expression of an infinite series is rather to be regarded as the infinite expression."
§ 565
According to Hegel, Spinoza defines the infinite as the "absolute affirmation of any kind of natural existence" and the finite as "a determinateness, as a negation." Because the affirmation of natural existence is absolute, and not a negation by something else, it must be taken only in relation to itself and as independent of anything else. The finite, however, is "negation, a ceasing-to-be in the form of a relation to an other which begins outside it."
But, says Hegel, the absolute affirmation of an existence does not exhaust the notion of infinity, because even though it is a necessary condition for infinity, it is not alone a sufficient condition. Something is only infinity if its limits are negated, and a limit is a negation. Hence to negate a limit is to negate a negation, and thus the infinite is not something immediate but rather is something brought about through mediation.
And yet, Spinoza conceives the infinite and absolutely unified substance as having
the form of an inert unity, i.e. of a unity which is not self-mediated, of a fixity or rigidity in which the Notion of the negative unity of the self, i.e. subjectivity, is still lacking.
§ 566
The mathematical example with which he illustrates the true infinite is a space between two unequal circles which are not concentric, one of which lies inside the other without touching it.
Hegel says that Spinoza thought highly of this figure and the concept it illustrates, because he made it "the motto of his Ethics." Hegel quotes:
Mathematicians conclude', he says, 'that the inequalities possible in such a space are infinite, not from the infinite amount of parts, for its size is fixed and limited and 1 can assume larger and smaller such spaces, but because the nature of the fact surpasses every determinateness.'
Hegel claims that Spinoza rejects "that conception of the infinite which represents it as an amount or as a series which is not completed;" for Spinoza notes that the infinite is not beyond the space of the figures, but rather is "present and complete:"
this space is bounded, but it is infinite 'because the nature of the fact surpasses every determinateness', because the determination of magnitude contained in it cannot at the same time be represented as a quantum, or in Kant's words already quoted, the synthesis cannot be completed to form a (discrete) quantum.
(Hegel here mentions that in a later Remark he will explain how the opposition of continuous and discrete quantum leads to the infinite). Spinoza calls the infinite of a series, says Hegel, " the infinite of the imagination." On the other hand, Spinoza calls the infinite as a self-relation the "infinite of thought, or infinitum actu." That is, it is actually infinite, because "it is complete and present within itself." Thus Spinoza would call Hegel's examples of the series 0.285714 ... or 1 + a + a2 + a3 ... infinities of the imagination or supposition, because they have no actuality on account of each one lacking something. Yet on the other hand, Spinoza would consider the 2/7 or 1/(1 - a) as actual infinities, because they are like the finite magnitude of enclosed space between the circles; for, these infinities contain within them the negation of the negation which makes them unlimited despite having a value that is represented fully in its formulation.
The 2/7 or 1/(1 - a) is equally a finite magnitude like Spinoza's space enclosed between two circles, with its inequalities, and can like this space be made larger or smaller.
And yet, Spinoza's actual infinite does involve "the absurdity of a larger or smaller infinite," because these formulations do not try to give a determinate value for them, so as indeterminate already we might consider them as having instead been larger or smaller without absurdity. Instead, these figures represent an infinite series that equals a finite "deficient" quantum. Our imaginations are only able to picture a finite quantum, because they cannot reflect on the "qualitative relation which constitutes the ground of the existing incommensurability."
From the original text:
Quantum
Remark 1: The Specific Nature of the Notion of the Mathematical Infinite
§ 538
The mathematical infinite has a twofold interest. On the one hand its introduction into mathematics has led to an expansion of the science and to important results; but on the other hand it is remarkable that mathematics has not yet succeeded in justifying its use of this infinite by the Notion (Notion taken in its proper meaning). Ultimately, the justifications are based on the correctness of the results obtained with the aid of the said infinite, which correctness is proved on quite other grounds: but the justifications are not based on the clarity of the subject matter and on the operation through which the results are obtained, for it is even admitted that the operation itself is incorrect.
This alone is in itself a bad state of affairs; such a procedure is unscientific. But it also involves the drawback that mathematics, being unaware of the nature of this its instrument because it has not mastered the metaphysics and critique of the infinite, is unable to determine the scope of its application and to secure itself against the misuse of it.
But in a philosophical respect the mathematical infinite is important because underlying it, in fact, is the notion of the genuine infinite and it is far superior to the ordinary so-called metaphysical infinite on which are based the objections to the mathematical infinite. Often, the science of mathematics can only defend itself against these objections by denying the competence of metaphysics, asserting that it has nothing to do with that science and does not have to trouble itself about metaphysical concepts so long as it operates consistently within its own sphere. Mathematics has to consider not what is true in itself but what is true in its own domain. Metaphysics, though disagreeing with the use of the mathematical infinite, cannot deny or invalidate the brilliant results obtained from it, and mathematics cannot reach clearness about the metaphysics of its own concept or, therefore, about the derivation of the modes of procedure necessitated by the use of the infinite.
If it were solely the difficulty of the Notion as such which troubled mathematics, it could ignore it without more ado since the Notion is more than merely the statement of the essential determinatenesses of a thing, that is, of the determinations of the understanding: and mathematics has seen to it that these determinatenesses are not lacking in precision; for it is not a science which has to concern itself with the concepts of its objects and which has to generate their content by explicating the concept, even if this could be effected only by ratiocination. But mathematics, in the method of its infinite, finds a radical contradiction to that very method which is peculiar to itself and on which as a science it rests. For the infinitesimal calculus permits and requires modes of procedure which mathematics must wholly reject when operating with finite quantities, and at the same time it treats these infinite quantities as if they were finite and insists on applying to the former the same modes of operation which are valid for the latter; it is a cardinal feature in the development of this science that it has succeeded in applying to transcendental determinations and their treatment the form of ordinary calculation.
Mathematics shows that, in spite of the clash between its modes of procedure, results obtained by the use of the infinite completely agree with those found by the strictly mathematical, namely, geometrical and analytical method. But in the first place, this does not apply to every result and the introduction of the infinite is not for the sole purpose of shortening the ordinary method but in order to obtain results which this method is unable to secure. Secondly, success does not by itself justify the mode of procedure. This procedure of the infinitesimal calculus shows itself burdened with a seeming inexactitude, namely, having increased finite magnitudes by an infinitely small quantity, this quantity is in the subsequent operation in part retained and in part ignored. The peculiarity of this procedure is that in spite of the admitted inexactitude, a result is obtained which is not merely fairly close and such that the difference can be ignored, but is perfectly exact. In the operation itself, however, which precedes the result, one cannot dispense with the conception that a quantity is not equal to nothing, yet is so inconsiderable that it can be left out of account. However, what is to be understood by mathematical determinateness altogether rules out any distinction of a greater or lesser degree of exactitude, just as in philosophy there can be no question of greater or less probability but solely of Truth. Even if the method and use of the infinite is justified by the result, it is nevertheless not so superfluous to demand its justification as it seems in the case of the nose to ask for a proof of the right to use it. For mathematical knowledge is scientific knowledge, so that the proof is essential; and even with respect to results it is a fact that a rigorous mathematical method does not stamp all of them with the mark of success, which in any case is only external.
It is worth while considering more closely the mathematical concept of the infinite together with the most noteworthy of the attempts aimed at justifying its use and eliminating the difficulty with which the method feels itself burdened. The consideration of these justifications and characteristics of the mathematical infinite which I shall undertake at some length in this Remark will at the same time throw the best light on the nature of the true Notion itself and show how this latter was vaguely present as a basis for those procedures.
The usual definition of the mathematical infinite is that it is a magnitude than which there is no greater (when it is defined as the infinitely great) or no smaller (when it is defined as the infinitely small), or in the former case is greater than, in the latter case smaller than, any given magnitude. It is true that in this definition the true Notion is not expressed but only, as already remarked, the same contradiction which is present in the infinite progress; but let us see what is implicitly contained in it. In mathematics a magnitude is defined as that which can be increased or diminished; in general, as an indifferent limit. Now since the infinitely great or small is that which cannot be increased or diminished, it is in fact no longer a quantum as such.
This is a necessary and direct consequence. But it is just the reflection that quantum (and in this remark quantum as such, as we find it, I call finite quantum) is sublated, which is usually not made, and which creates the difficulty for ordinary thinking; for quantum in so far as it is infinite is required to be thought as sublated, as something which is not a quantum but yet retains its quantitative character.
To quote Kant's opinion of the said definition which he finds does not accord with what is understood by an infinite whole: 'According to the usual concept, a magnitude is infinite beyond which there can be no greater (i.e. greater than the amount contained therein of a given unit); but there can be no greatest amount because one or more units can always be added to it. But our concept of an infinite whole does not represent how great it is and it is not therefore the concept of a maximum (or a minimum); this concept rather expresses only the relation of the whole to an arbitrarily assumed unit, with respect to which the relation is greater than any number. According as this assumed unit is greater or smaller, the infinite would be greater or smaller. The infinity, however, since it consists solely in the relationto this given unit, would always remain the same, although of course the absolute magnitude of the whole would not be known through it.'
Kant objects to infinite wholes being regarded as a maximum, as a completed amount of a given unit. The maximum or minimum as such still appears as a quantum, an amount. Such a conception cannot avert the conclusion, adduced by Kant, which leads to a greater or lesser infinite. And in general, so long as the infinite is represented as a quantum, the distinction of greater or less still applies to it. This criticism does not however apply to the Notion of the genuine mathematical infinite, of the infinite difference, for this is no longer a finite quantum.
Kant's concept of infinite, on the other hand, which he calls truly transcendental is 'that the successive synthesis of the unit in the measurement of a quantum can never be completed'. A quantum as such is presupposed as given; by synthesising the unit this is supposed to be converted into an amount, into a definite assignable quantum; but this synthesis, it is said, can never be completed. It is evident from this that we have here nothing but an expression of the progress to infinity, only represented transcendentally, i.e. properly speaking, subjectively and psychologically. True, in itself the quantum is supposed to be completed; but transcendentally, namely in the subject which gives it a relationto a unit, the quantum comes to be determined only as incomplete and as simply burdened with a beyond. Here, therefore, there is no advance beyond the contradiction contained in quantity; but the contradiction is distributed between the object and the subject, limitedness being ascribed to the former, and to the latter the progress to infinity, in its spurious sense, beyond every assigned determinateness.
On the other hand, it was said above that the character of the mathematical infinite and the way it is used in higher analysis corresponds to the Notion of the genuine infinite; the comparison of the two determinations will now be developed in detail. In the first place, as regards the true infinite quantum, it was characterised as in its own selfinfinite; it is such because, as we have seen, the finite quantum or quantum as such and its beyond, the spurious infinite, are equally sublated. Thus the sublated quantum has returned into a simple unity and self-relation; but not merely like the extensive quantum which, in passing into intensive quantum, has its determinateness only in itself [or implicitly] inan external plurality, towards which, however, it is indifferent and from which it is supposed to be distinct.
The infinite quantum, on the contrary, contains within itself first externality and secondly the negation of it; it is thus no longer any finite quantum, not a quantitative determinateness which would have a determinate being as quantum; it is simple, and therefore only a moment. It is a quantitative determinateness in qualitative form; its infinity consists in its being a qualitative determinateness. As such moment, it is in essential unity with its other, and is only as determined by this its other, i.e. it has meaning solely with reference to that which stands in relation to it. Apart from this relation it is a nullity — simply because quantum as such is indifferent to the relation, yet in the relation is supposed to be an immediate, inert determination. As only a moment, it is, in the relation, not an independent, indifferent something; the quantum in its infinity is a being-for-self, for it is at the same time a quantitative determinateness only in the form of a being-for-one.
The Notion of the infinite as abstractly expounded here will show itself to be the basis of the mathematical infinite and the Notion itself will become clearer if we consider the various stages in the expression of a quantum as moment of a ratio, from the lowest where it is still also a quantum as such, to the higher where it acquires the meaning and the expression of a properly infinite magnitude.
Let us then first take quantum in the relation where it is a fractional number. Such fraction, 2/7 for example, is not a quantum like 1, 2, 3, etc.; although it is an ordinary finite number it is not an immediate one like the whole numbers but, as a fraction, is directly determined by two other numbers which are related to each other as amount and unit, the unit itself being a specific amount. However, if we abstract from this more precise determination of them and consider them solely as quanta in the qualitative relation in which they are here, then 2 and 7 are indifferent quanta; but since they appear here only as moments, the one of the other, and consequently of a third (of the quantum which is called the exponent), they directly count no longer simply as 2 and 7 but only according to the specific relationship in which they stand to each other. In their place, therefore, we can just as well put 4 and 14, or 6 and 21, and so on to infinity. With this, then, they begin to have a qualitative character. If 2 and 7 counted as mere quanta, then 2 is just 2 and nothing more, and 7 is simply 7; 4, 14, 6, 21 etc., are completely different from them and, as only immediate quanta, cannot be substituted for them. But in so far as 2 and 7 are not to be taken as such immediate quanta their indifferent limit is sublated; on this side therefore they contain the moment of infinity, since not only are they no longer merely 2 and 7, but their quantitative determinateness remains — but as one which is in itself qualitative, namely in accordance with their significance as moments in the ratio. Their place can be taken by infinitely many others without the value of the fraction being altered, by virtue of the determinateness possessed by the ratio.
However, the representation of infinity by a fractional number is still imperfect because the two sides of the fraction, 2 and 7, can be taken out of the relation and are ordinary, indifferent quanta; their connection as moments of the ratio is an external circumstance which does not directly concern them. Their relation, too, is itself an ordinary quantum, the exponent of the ratio.
The letters with which general arithmetic operates, the next universality into which numbers are raised, do not possess the property of having a specific numerical value; they are only general symbols and indeterminate possibilities of any specific value. The fraction a/b seems, therefore, to be a more suitable expression of the infinite, since a and b, taken out of their relation to each other, remain undetermined, and taken separately, too, have no special peculiar value. However, although these letters are posited as indeterminate magnitudes their meaning is to be some finite quantum. Therefore, though they are the general representation of number, it is only of a determinate number, and the fact that they are in a ratio is likewise an inessential circumstance and they retain their value outside it.
If we consider more closely what is present in the ratio we find that it contains the following two determinations: first it is a quantum, secondly, however, this quantum is not immediate but contains qualitative opposition; at the same time it remains therein a determinate, indifferent quantum by virtue of the fact that it returns into itself from its otherness, from the opposition, and so also is infinite. These two determinations are represented in the following familiar form developed in their difference from each other.
The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as anaggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series; but they can all be so expressed in a numerical system based on another unit.
Now in the infinite series, which is supposed to represent the fraction as an amount, the aspect of the fraction as a ratio has vanished and with it there has vanished too the aspect which, as we have already shown, makes the fraction in its own self infinite. But this infinity has entered in another way; the series, namely, is itself infinite.
Now the nature of this infinity of the series is self-evident; it is the spurious infinity of the progression. The series contains and exhibits the contradiction of representing that which is a relation possessing a qualitative nature, as devoid of relation, as a mere quantum, as an amount. The consequence of this is that the amount which is expressed in the series always lacks something, so that in order to reach the required determinateness, we must always go further than the terms already posited. The law of the progression is known, it is implicit in the determination of the quantum contained in the fraction and in the nature of the form in which it is supposed to be expressed. By continuing the series the amount can of course be made as accurate as required; but representation by means of the series continues to remain only an ought-to-be; it is burdened with a beyond which cannot be sublated, because to express as an amount that which rests on a qualitative determinateness is a lasting contradiction.
In this infinite series, this inexactitude is actually present, whereas in the genuine mathematical infinite there is only an appearance of inexactitude. These two kinds of mathematical infinite are as little to be confounded as are the two kinds of philosophical infinite. In representing the genuine mathematical infinite, the form of series was used originally and it has recently again been invoked; but this form is not necessary for it. On the contrary, the infinite of the infinite series is essentially different from the genuine infinite as the sequel will show. Indeed the form of infinite series is even inferior to the fractional expression.
For the infinite series contains the spurious infinity, because what the series is meant to express remains an ought-to-be and what it does express is burdened with a beyond which does not vanish and differs from what was meant to be expressed. It is infinite not because of the terms actually expressed but because they are incomplete, because the otherwhich essentially belongs to them is beyond them; what is really present in the series, no matter how many terms there may be, is only something finite, in the proper meaning of that word, posited as finite, i.e., as something which is not what it ought to be. But on the other hand, what is called the finite expression or the sum of such a series lacks nothing; it contains that complete value which the series only seeks; the beyond is recalled from its flight; what it is and what it ought to be are not separate but the same.
What distinguishes these two is more precisely this, that in the infinite series the negative is outside its terms which are present only qua parts of the amount. On the other hand, in the finite expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of the ratio and this is an accomplished return-into-self, a self-related unity as a negation of the negation (both sides of the ratio are only moments), and consequently has within itthe determination of infinity. Thus the usually so-called sum, the 2/7 or 1/(1 - a) is in fact a ratio; and this so-calledfinite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate. Furthermore, the series is in fact the finite expression; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be; and then, too, what it lacks is itself a specific quantum; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted; this is a kind of superstition, the superstition of the understanding; we have seen how, on the contrary, it indicates only a deficiency.
We may further remark that the existence of infinite series which cannot be summed is an external and contingent circumstance with respect to the form of series as such. They contain a higher kind of infinity than do those which can be summed, namely an incommensurability, or the impossibility of representing the quantitative ratio contained in them as a quantum, even in the form of a fraction; but the form of series as such which they have contains the same determination of spurious infinity that is present in the series capable of summation.
The terminological inversion we have just noticed in connection with the fraction and its expression as a series, also occurs when the mathematical infinite — not the one just named but the genuine infinite — is called the relativeinfinite, while the ordinary metaphysical — by which is understood the abstract, spurious infinite is called absolute.But in point of fact it is this metaphysical infinite which is merely relative, because the negation which it expresses is opposed to a limit only in such a manner that this limit persists outside it and is not sublated by it; the mathematical infinite, on the contrary, has within itself truly sublated the finite limit because the beyond of the latter is united with it.
It is primarily in this sense, in which it has been demonstrated that the so-called sum or finite expression of an infinite series is rather to be regarded as the infinite expression, that Spinoza opposes the concept of true infinity to that of the spurious and illustrates it by examples. It will shed most light on his concept if I follow up this exposition with what he says on the subject.
He starts by defining the infinite as the absolute affirmation of any kind of natural existence, the finite on the contrary as a determinateness, as a negation. That is to say, the absolute affirmation of an existence is to be taken as its relation to itself, its not being dependent on an other; the finite, on the other hand, is negation, a ceasing-to-be in the form of arelation to an other which begins outside it. Now the absolute affirmation of an existence does not, it is true, exhaust the notion of infinity; this implies that infinity is an affirmation, not as immediate, but only as restored by the reflection of the other into itself, or as negation of the negative. But with Spinoza, substance and its absolute unity has the form of an inert unity, i.e. of a unity which is not self-mediated, of a fixity or rigidity in which the Notion of the negative unity of the self, i.e. subjectivity, is still lacking.
The mathematical example with which he illustrates the true infinite is a space between two unequal circles which are not concentric, one of which lies inside the other without touching it. It seems that he thought highly of this figure and of the concept which it was used to illustrate, making it the motto of his Ethics. 'Mathematicians conclude', he says, 'that the inequalities possible in such a space are infinite, not from the infinite amount of parts, for its size is fixed and limited and 1 can assume larger and smaller such spaces, but because the nature of the fact surpasses every determinateness.' It is evident that Spinoza rejects that conception of the infinite which represents it as an amount or as a series which is not completed, and he points out that here, in the space of his example, the infinite is not beyond, but actually present and complete; this space is bounded, but it is infinite 'because the nature of the fact surpasses every determinateness', because the determination of magnitude contained in it cannot at the same time be represented as a quantum, or in Kant's words already quoted, the synthesis cannot be completed to form a (discrete) quantum. How in general the opposition of continuous and discrete quantum leads to the infinite, will be shown in detail in a later Remark. Spinoza calls the infinite of a series the infinite of the imagination; on the other hand, the infinite as self-relation he calls the infinite of thought, or infinitum actu. It is, namely, actu, actually infinite because it is complete and present within itself. Thus the series 0.285714 ... or 1 + a + a2 + a3 ... is the infinite merely of imagination or supposition; for it has no actuality, it definitely lacks something; on the other hand 2/7 or 1/(1 - a) is actually not only what the series is in its developed terms, but is, in addition, what the series lacks, what it only ought to be. The 2/7 or 1/(1 - a) is equally a finite magnitude like Spinoza's space enclosed between two circles, with its inequalities, and can like this space be made larger or smaller. But this does not involve the absurdity of a larger or smaller infinite; for this quantum of the whole does not concern the relation of its moments, the nature of the fact, i.e. the qualitative determination of magnitude; what is actually present in the infinite series is equally a finite quantum, but it is also still deficient. Imagination on the contrary stops short at quantum as such and does not reflect on the qualitative relation which constitutes the ground of the existing incommensurability.
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