Hegel’s Science of Logic
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.
§ 539
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.
§ 540
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.
§ 541
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.
§ 542
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.
§ 543
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.
§ 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. 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.
§ 545
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.
§ 546
To quote Kant's opinion of the said definition which he finds does not accord with what is understood by an infinite whole: 'According to the usual concept, a magnitude is infinite beyond which there can be no greater (i.e. greater than the amount contained therein of a given unit); but there can be no greatest amount because one or more units can always be added to it. But our concept of an infinite whole does not represent how great it is and it is not therefore the concept of a maximum (or a minimum); this concept rather expresses only the relation of the whole to an arbitrarily assumed unit, with respect to which the relation is greater than any number. According as this assumed unit is greater or smaller, the infinite would be greater or smaller. The infinity, however, since it consists solely in the relation to this given unit, would always remain the same, although of course the absolute magnitude of the whole would not be known through it.'
§ 547
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.
§ 548
Kant's concept of infinite, on the other hand, which he calls truly transcendental is 'that the successive synthesis of the unit in the measurement of a quantum can never be completed'. A quantum as such is presupposed as given; by synthesising the unit this is supposed to be converted into an amount, into a definite assignable quantum; but this synthesis, it is said, can never be completed. It is evident from this that we have here nothing but an expression of the progress to infinity, only represented transcendentally, i.e. properly speaking, subjectively and psychologically. True, in itself the quantum is supposed to be completed; but transcendentally, namely in the subject which gives it a relation to a unit, the quantum comes to be determined only as incomplete and as simply burdened with a beyond. Here, therefore, there is no advance beyond the contradiction contained in quantity; but the contradiction is distributed between the object and the subject, limitedness being ascribed to the former, and to the latter the progress to infinity, in its spurious sense, beyond every assigned determinateness.
§ 549
On the other hand, it was said above that the character of the mathematical infinite and the way it is used in higher analysis corresponds to the Notion of the genuine infinite; the comparison of the two determinations will now be developed in detail. In the first place, as regards the true infinite quantum, it was characterised as in its own self infinite; it is such because, as we have seen, the finite quantum or quantum as such and its beyond, the spurious infinite, are equally sublated. Thus the sublated quantum has returned into a simple unity and self-relation; but not merely like the extensive quantum which, in passing into intensive quantum, has its determinateness only in itself [or implicitly] in an external plurality, towards which, however, it is indifferent and from which it is supposed to be distinct.
§ 550
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.
§ 551
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.
§ 552
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.
§ 553
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.
§ 554
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.
§ 555
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.
§ 556
The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as an aggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series; but they can all be so expressed in a numerical system based on another unit.
§ 557
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.
§ 558
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.
§ 559
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.
§ 560
For the infinite series contains the spurious infinity, because what the series is meant to express remains an ought-to-be and what it does express is burdened with a beyond which does not vanish and differs from what was meant to be expressed. It is infinite not because of the terms actually expressed but because they are incomplete, because the other which essentially belongs to them is beyond them; what is really present in the series, no matter how many terms there may be, is only something finite, in the proper meaning of that word, posited as finite, i.e., as something which is not what it ought to be. But on the other hand, what is called the finite expression or the sum of such a series lacks nothing; it contains that complete value which the series only seeks; the beyond is recalled from its flight; what it is and what it ought to be are not separate but the same.
§ 561
What distinguishes these two is more precisely this, that in the infinite series the negative is outside its terms which are present only qua parts of the amount. On the other hand, in the finite expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of the ratio and this is an accomplished return-into-self, a self-related unity as a negation of the negation (both sides of the ratio are only moments), and consequently has within it the determination of infinity. Thus the usually so-called sum, the 2/7 or 1/(1 - a) is in fact a ratio; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate. Furthermore, the series is in fact the finite expression; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be; and then, too, what it lacks is itself a specific quantum; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted; this is a kind of superstition, the superstition of the understanding; we have seen how, on the contrary, it indicates only a deficiency.
§ 562
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.
§ 563
The terminological inversion we have just noticed in connection with the fraction and its expression as a series, also occurs when the mathematical infinite — not the one just named but the genuine infinite — is called the relative infinite, while the ordinary metaphysical — by which is understood the abstract, spurious infinite is called absolute. But in point of fact it is this metaphysical infinite which is merely relative, because the negation which it expresses is opposed to a limit only in such a manner that this limit persists outside it and is not sublated by it; the mathematical infinite, on the contrary, has within itself truly sublated the finite limit because the beyond of the latter is united with it.
§ 564
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.
§ 565
He starts by defining the infinite as the absolute affirmation of any kind of natural existence, the finite on the contrary as a determinateness, as a negation. That is to say, the absolute affirmation of an existence is to be taken as its relation to itself, its not being dependent on an other; the finite, on the other hand, is negation, a ceasing-to-be in the form of a relation to an other which begins outside it. Now the absolute affirmation of an existence does not, it is true, exhaust the notion of infinity; this implies that infinity is an affirmation, not as immediate, but only as restored by the reflection of the other into itself, or as negation of the negative. But with Spinoza, substance and its absolute unity has the form of an inert unity, i.e. of a unity which is not self-mediated, of a fixity or rigidity in which the Notion of the negative unity of the self, i.e. subjectivity, is still lacking.
§ 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. 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.
§ 567
The incommensurability which lies in Spinoza's example embraces in general the functions of curved lines and more precisely, leads to the infinite which mathematics has introduced with such functions, in general, with the functions of variable magnitudes. This infinite is the genuine mathematical qualitative infinite which Spinoza also had in mind. We shall now consider this determination here in detail.
§ 568
First of all, as regards the category of variability which is accorded such importance and which embraces the magnitudes related in these functions, it is to be noted that these magnitudes are not supposed to be variable in the way that the two numbers 2 and 7 are in the fraction 2/7: their place can equally well be taken by 4 and 14, 6 and 21, and by other numbers ad infinitum without altering the value of the fraction; and still more in a/b, can a and b be replaced by any arbitrary number without altering what a/b is intended to express. Now in the sense that in the place, too, of x and y of a function, there can be put an infinite, i.e. inexhaustible, multitude of numbers, a and b are just as much variable magnitudes as the said x and y. The expression 'variable magnitudes' is therefore very vague and ill-chosen for those determinations of magnitude whose interest and manner of treatment lie in something quite distinct from their mere variability.
§ 569
In order to make clear wherein lies the true character of those moments of a function with which higher analysis is concerned, we must once more run through the stages to which we have already drawn attention. In 2/7 or a/b, 2 and 7 are each independent determinate quanta and the relation is not essential to them; a and b likewise are intended to represent quanta which remain what they are even outside the relation. And further, 2/7 and a/b are each a fixed quantum, a quotient; the ratio constitutes an amount, the unit of which is expressed by the denominator and the amount of these units by the numerator, or conversely; even if 4 and 14, and so on, are substituted for 2 and 7, the ratio, also as a quantum, remains the same. But now in the function y2/x = p, for example, this is essentially changed; here, it is true that x and y can stand for definite quanta, but only x and y2 have a determinate quotient, not x and y. Hence not only are these sides of the ratio x and y, not any determinate quanta, but, secondly, their ratio is not a fixed quantum (nor is such a quantum meant as in the case of a and b), not a fixed quotient, but this quotient is, as a quantum, absolutely variable. But this is solely because x has a relation, not to y, but to the square of y. The relation of a magnitude to a power is not a quantum, but essentially a qualitative relation; the power-relation is the feature which is to be regarded as the fundamental determination. But in the function of the straight line y = ax, a is an ordinary fraction and quotient; consequently this function is only formally a function of variable magnitudes, or x and y here are what a and b are in a/b that is, they are not in that determination in which the differential and integral calculus considers them. On account of the special nature of the variable magnitudes in this mode of consideration, it would have been fitting to have introduced both a special name for them and other symbols than those generally used for unknown quantities in any finite equation, determinate or indeterminate; for there is an essential difference between those magnitudes and such quanta which are merely unknown, but are in themselves completely determined or are a definite range of determinate quanta. It is, too, only because of a lack of awareness of what constitutes the peculiar interest of higher analysis and of what has led to the need for and invention of the differential calculus, that functions of the first degree and the equation of the straight line are themselves included in the treatment of this calculus; such formalism originates partly, too, in the mistake of imagining that the intrinsically correct demand for the generalisation of a method has been fulfilled when the specific determinateness on which the need for the calculus is based is omitted, as if in this domain we were concerned only with variable magnitudes. A great deal of formalism would, indeed, have been avoided if it had been perceived that the calculus is concerned not with variable magnitudes as such but with relations of powers.
§ 570
But there is still another stage where the peculiar character of the mathematical infinite becomes prominent. In an equation in which x and y are determined primarily by a power-relation, x and y as such are still supposed to signify quanta; now this significance is altogether and completely lost in the so-called infinitesimal differences. Dx, dy, are no longer quanta, nor are they supposed to signify quanta; it is solely in their relation to each other that they have any meaning, a meaning merely as moments. They are no longer something (something taken as a quantum), not finite differences; but neither are they nothing; not empty nullities. Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as determinations of the differential coefficient dx/dy.
§ 571
In this concept of the infinite, the quantum is genuinely completed into a qualitative reality; it is posited as actually infinite; it is sublated not merely as this or that quantum but as quantum generally. But the quantitative determinateness remains as element of the principle of the quanta, or, as has also been said, the quanta remain in their first concept.
§ 572
It is this concept which has been the target for all the attacks made on the fundamental determination of the mathematics of this infinite, i.e. of the differential and integral calculus. Failure to recognise it was the result of incorrect ideas on the part of mathematicians themselves; but it is the inability to justify the object as Notion which is mainly responsible for these attacks. But mathematics, as we remarked above, cannot evade the Notion here; for, as mathematics of the infinite, it does not confine itself to the finite determinateness of its objects (as in ordinary mathematics, which considers and relates space and number and their determinations only according to their finitude); on the contrary, when it treats a determination taken from ordinary mathematics, it converts it into an identity with its opposite, e.g. converting a curved line into a straight one, the circle into a polygon, etc. Consequently, the operations which it allows itself to perform in the differential and integral calculus are in complete contradiction with the nature of merely finite determinations and their relations and would therefore have to be justified solely by the Notion.
§ 573
Although the mathematics of the infinite maintained that these quantitative determinations are vanishing magnitudes, i.e. magnitudes which are no longer any particular quantum and yet are not nothing but are still a determinateness relatively to an other, it seemed perfectly clear that such an intermediate state, as it was called, between being and nothing does not exist. What we are to think of this objection and the so-called intermediate state, has already been indicated above in Remark 4 to the category of becoming. The unity of being and nothing is, of course, not a state; a state would be a determination of being and nothing into which these moments might be supposed to have lapsed only by accident, as it were, into a diseased condition externally induced through erroneous thinking; on the contrary, this mean and unity, the vanishing or equally the becoming is alone their truth.
§ 574
Further, it has been said that what is infinite is not comparable as something greater or smaller; therefore there cannot be a relation between infinites according to orders or dignities of the infinite, although in the science of infinitesimals these distinctions do occur. Underlying this objection already mentioned is always the idea that here we are supposed to be dealing with quanta which are compared as quanta, that determinations which are no longer quanta no longer have any relationship to each other. But the truth is rather that that which has being solely in the ratio is not a quantum; the nature of quantum is such that it is supposed to have a completely indifferent existence apart from its ratio, and its difference from another quantum is supposed not to concern its own determination; on the other hand the qualitative is what it is only in its distinction from an other. The said infinite magnitudes, therefore, are not merely comparable, but they exist only as moments of comparison, i.e. of the ratio.
§ 575
I will adduce the most important definitions of this infinite which have been given in mathematics. From these it will be clear that the thought underlying them accords with the Notion developed here, but that the originators of the definitions did not establish the thought as Notion and found it necessary in the application to resort again to expedients which conflict with their better cause.
§ 576
The thought cannot be more correctly determined than in the way Newton has stated it. I eliminate here those determinations which belong to the idea of motion and velocity (from which, mainly, he took the name of fluxions) because in them the thought does not appear in its proper abstraction but as concrete and mixed with non-essential forms. Newton explains that he understands by these fluxions not indivisibles (a form which was used by earlier mathematicians, Cavalieri and others and which involves the concept of an intrinsically determinate quantum), but vanishing divisibles; also not sums and ratios of determinate parts but the limits (limites) of sums and ratios. It may be objected that vanishing magnitudes do not have a final ratio, because the ratio before it vanishes is not final, and when it has vanished is no longer a ratio. But by the ratio of vanishing magnitudes is to be understood not the ratio before which and after which they vanish, but with which they vanish. (quacum evanescunt). Similarly, the first ratio of nascent magnitudes is that with which they become.
§ 577
Newton did what the scientific method of his time demanded, he only explained what was to be understood by an expression; but that such and such is to be understood by it is, properly speaking, a subjective presumption, or a historical demand, without any indication that such a concept is in itself absolutely necessary or that there is truth in it. However, what has been quoted shows that the concept put forward by Newton corresponds to the way in which infinite quantity resulted from the reflection of quantum into itself in the exposition above. By magnitudes is understood magnitudes in their vanishing, i.e. which are no longer quanta; also, not ratios of determinate parts, but the limits of the ratio. The meaning is, therefore, that with the vanishing of the quanta individually, the sides of the ratio, there also vanishes the ratio itself in so far as it is a quantum; the limit of the quantitative ratio is that in which it both is and is not, or, more precisely, in which the quantum has vanished, with the result that the ratio and its sides are preserved, the former only as a qualitative relation of quantity and the latter similarly as qualitative moments of quantity. Newton goes on to add that from the fact that there are final ratios of vanishing magnitudes, it must not be inferred that there are final magnitudes, indivisibles. For this would mean a leap again from the abstract ratio to its sides as supposedly having an independent value of their own as indivisibles outside their relation, as something which would be a one, something devoid of any relation at all.
§ 578
To prevent such a misunderstanding, he again points out that final ratios are not ratios of final magnitudes, but are limits to which the ratios of the magnitudes decreasing without limit are nearer than any given, i.e. finite, difference; the ratios, however, do not exceed these limits, for if they did they would become nullities. In other words, final magnitudes could have been taken to mean, as already said, indivisibles or ones. But the definition of the final ratio excludes the conception both of the indifferent one which is devoid of any relation, and of the finite quantum. If the required determination had been developed into the Notion of a quantitative determination which is purely a moment of the ratio, there would have been no need for the decreasing without limit into which Newton converts the quantum and which only expresses the progress to infinity, or for the determination of divisibility which no longer has any immediate meaning here.
§ 579
As regards the preservation of the ratio in the vanishing of the quanta, there is found elsewhere, as in Carnot, the expression that by virtue of the law of continuity, the vanishing magnitudes still retain the ratio from which they come, before they vanish.
§ 580
This conception expresses the true nature of the matter, if the continuity of the quantum is not understood to be the continuity which it has in the infinite progress where the quantum is continued in its vanishing in such a manner that in its beyond there arises only a finite quantum again, only a fresh term of the series; but a continuous progress is always imagined as one in which values are passed through, values which are still finite quanta. On the other hand, where the transition is made into the true infinite it is the ratio that is continuous; so continuous is it, so completely is it preserved, that the transition may be said to consist solely in throwing into relief the pure ratio and causing the non-relational determination — i.e. that a quantum which is a side of the ratio is still a quantum outside this relation — to vanish. This purification of the quantitative ratio is thus analogous to grasping an empirical reality in terms of its Notion. The empirical reality is thereby raised above itself in such a way that its Notion contains the same characteristic features as it has itself, but these are grasped in their essentiality and are taken into the unity of the Notion in which they have lost their indifferent, Notion — less existence.
§ 581
The other form of Newton's exposition of the magnitudes in question is equally interesting, namely, as generative magnitudes or principles. A generated magnitude (genita) is a product or quotient, such as a root, rectangle, square, also the sides of rectangles and squares — in general, a finite magnitude. 'Such a magnitude being considered as variable, increasing or decreasing in ceaseless motion and flux, he gives its momentary increments or decrements the name of moments. But these are not to be taken for particles of a definite magnitude (particulae finitae): such would not themselves be moments but magnitudes generated from moments. Rather are they to be understood as the nascent principles or beginnings of finite magnitudes.' Here the quantum is distinguished from itself: as a product or a real being [Daseiendes], and in its becoming (or as nascent), in its beginning and principle, that is to say, in its Notion, or, what is here the same thing, in its qualitative determination: in the latter the quantitative differences, the infinite increments or decrements, are only moments; only that which has becoming at its back has passed over into the indifference of determinate being and into externality, i.e. is quantum. But if on the one hand the philosophy of the true Notion must acknowledge these determinations of the infinite with respect to increments or decrements, on the other hand it must be observed that the very forms of increments etc. fall within the category of immediate quantum and of the continuous progress to which we have referred; in fact the conceptions of increment, growth or increase of x by dx or i, and so on, are to be regarded as the fundamental vice in these methods — the permanent obstacle to disengaging the determination of the qualitative moment of quantity in its purity from the conception of the ordinary quantum.
§ 582
The conception of infinitesimals which is implicit, too, in the increment or decrement itself, is much inferior to the above determinations. The nature of these magnitudes is supposed to be such that they may be neglected, not only in comparison with finite magnitudes, but also their higher orders in comparison with their lower, and even the products of several in comparison with a single one. With Leibniz, this demand to neglect is more strikingly prominent than with previous inventors of methods relating to these infinitesimals in which this call to neglect also occurs. It is chiefly this call to neglect which, along with a gain in facility, has given this calculus the appearance of inexactitude and express incorrectness in its method of procedure. Wolf has tried to make this neglect intelligible in his own way of popularising things, i.e. by polluting the pure Notion and setting in its place incorrect sensuous conceptions. For example, he compares the neglect of infinitesimals of higher orders relatively to lower with the procedure of a surveyor who, in measuring the height of a mountain is no less accurate if meanwhile the wind has blown away a grain of sand from the top; or with the neglect of the height of houses or towers when calculating lunar eclipses.
§ 583
Even if ordinary common sense in fairness allows such inexactitude, all geometricians reject this conception. It is quite obvious that in the science of mathematics there cannot be any question of such empirical accuracy; mathematical measuring by operations of the calculus or by geometrical constructions and proofs is altogether different from land-surveying, from the measuring of empirical lines, figures etc. Besides, by comparing the result obtained by a strictly geometrical method with that obtained by the method of infinite differences, analysts demonstrate that the one is the same as the other and that there is absolutely no question of a greater or lesser degree of exactness. And it is self-evident that an absolutely exact result could not emerge from an inexact method. Yet on the other hand again, the method itself cannot do without this omission of what is regarded as insignificant, despite its protestations against the way this omission is justified. And this is the difficulty which engages the efforts of the analysts to make intelligible and to remove the inherent inconsistency.
§ 584
It is especially Euler's conception of the matter which must be cited here. He adopts the general Newtonian definition and insists that the differential calculus considers the ratios of the increments of a magnitude, but that the infinite difference as such is to be considered as wholly nil. How this is to be understood is clear from the foregoing; the infinite difference is a nil only of quantum, not a qualitative nil, but as a nil of quantum it is a pure moment of the ratio only. It is not a quantitative difference; but for that reason it is, on the one hand, altogether wrong to speak of those moments which are called infinitesimals, also as increments or decrements and as differences. This description implies that something is added to or subtracted from the initially given finite magnitude, that a subtraction or addition, an arithmetical, external operation takes place. But it is to be noticed that the transition of the function of the variable magnitude into its differential is of a quite different nature; as we have made clear, it is to be considered as a reduction of the finite function to the qualitative relation of its quantitative determinations. On the other hand, the error becomes obvious when it is said that the increments by themselves are zeros, that only their ratios are considered; for a zero no longer has any determinateness at all. This conception then, does get as far as the negative of the quantum and gives definite expression to it, but at the same time it does not grasp this negative in its positive significance of qualitative determinations of quantity which, if they were torn out of the ratio and regarded as quanta, would be only zeros.
§ 585
The opinion of Lagrange on the idea of limits or final ratios is that although one can well imagine the ratio of two magnitudes so long as they remain finite, this ratio does not present any clear and definite concept to the intellect as soon as its terms become simultaneously zero. And the understanding must, indeed, transcend this merely negative side on which the terms of the ratio are quantitatively zero, and must grasp them positively, as qualitative moments. But we cannot regard, as satisfactory Euler's further remarks with regard to this conception of his in which he tries to show that two so-called infinitesimals which are supposed to be nothing else but zeros, nevertheless stand in a relation to each other, for which reason they are denoted by symbols other than zero. He tries to base this on the difference between the arithmetical and geometrical ratio: in the former, we have an eye to the difference, in the latter, to the quotient, so that although in the former there is no difference between two zeros, this is not so in the geometrical ratio; if 2: 1 = 0 : 0 then from the nature of proportion it follows that, because the first term is twice as great as the second, the third is also twice as great as the fourth; thus according to proportion, 0 : 0 is to be taken as the ratio of 2 : 1. Even in common arithmetic n. 0 = 0 and therefore n : 1 = 0 : 0. But it is just because 2 : 1 or n : 1 is a relation of quanta that there cannot be any corresponding ratio or expression of 0 : 0.
§ 586
I refrain from citing any further instances since those already considered show clearly enough that the genuine Notion of the infinite is, in fact, implied in them, but that the specific nature of that Notion has not been brought to notice and grasped. Consequently, in the actual application of the method of infinitesimals, the genuine Notion of the infinite cannot exercise any influence; on the contrary, there is a return of the finite determinateness of quantity and the operation cannot dispense with the conception of a quantum which is merely relatively small. The calculus makes it necessary to subject the so-called infinitesimals to ordinary arithmetical operations of addition and so on, which are based on the nature of finite magnitudes, and therefore to regard them momentarily as finite magnitudes and to treat them as such. It is for the calculus to justify its procedure in which it first brings them down into this sphere and treats them as increments or differences, and then neglects them as quanta after it had just applied forms and laws of finite magnitudes to them.
I will proceed to cite the main features of the attempts of the geometricians to remove these difficulties.
§ 587
The older analysts had little scruples in the matter, but the moderns directed their efforts mainly towards bringing the differential calculus back to the evidence of a strictly geometrical method and in it to attain to the rigour of the proofs of the ancients (Lagrange's expressions) in mathematics. But since the principle of infinitesimal analysis is of a higher nature than the principle of the mathematics of finite magnitudes, that kind of evidence had perforce to be dispensed with, just as philosophy, too, cannot lay claim to that obviousness which belongs to the natural sciences, e.g. natural history — and just as eating and drinking are reckoned a more intelligible business than thinking and understanding. Accordingly, we shall deal only with the efforts to attain to the rigour of proof of the ancients.
§ 588
Some have attempted to dispense altogether with the concept of the infinite, and without it to achieve what seemed to be bound up with its use. Lagrange speaks, e.g., of the method devised by Landen, saying that it is purely analytical and does not employ infinitesimal differences, but starts with different values of variable magnitudes and subsequently equates them. He also gives it as his opinion that in this method, the differential calculus loses its own peculiar advantages, namely simplicity of method and facility of operation. This is, indeed, a procedure which in some measure corresponds to the starting-point of Descartes' tangential method of which detailed mention will be made later. This much, we may remark here, is generally evident, that the general procedure in which different values of variable magnitudes are assumed and subsequently equated, belongs to another department of mathematical treatment than that to which the method of the differential calculus itself belongs; and that the peculiar nature of the simple relation (to be considered in detail further on) to which its actual, concrete determination reduces, namely, of the derived function to the original, is not brought into prominence.
§ 589
The earlier of the moderns, Fermat, Barrow, and others for example, who at first used infinitesimals in that application which was subsequently developed into the differential and integral calculus, and then Leibniz, too, and those following him including Euler, always frankly believed that they were entitled to omit the products of infinitesimal differences and their higher powers, solely on the ground that they vanish relatively to the lower order. This is for them the sole basis of the fundamental principle, namely the determination of that which is the differential of a product or a power, for the entire theoretical teaching reduces to this. The rest is partly the mechanism of development and partly application, in which however as we shall later on see, the more important, or rather the sole, interest is to be found. With respect to the present topic, we need only mention here what is elementary, that on the same ground of insignificance, the cardinal principle adopted in relation to curves is that the elements of the curves, namely the increments of abscissa and ordinate, have the relation to each other of subtangent and ordinate; for the purpose of obtaining similar triangles, the arc which forms the third side of a triangle to the two increments of the characteristic triangle (as it rightly used to be called), is regarded as a straight line, as part of the tangent and one of the increments therefore as reaching to the tangent. By these assumptions those determinations are, on the one hand, raised above the nature of finite magnitudes, but on the other hand, a method which is valid only for finite magnitudes and which does not permit the omission of anything on the ground of insignificance, is applied to moments now called infinitesimal. With such a mode of procedure, the difficulty which encumbers the method remains in all its starkness.
§ 590
We must mention here a remarkable procedure of Newton the invention of an ingenious device to remove the arithmetically incorrect omission of the products of infinitesimal differences or higher orders of them in the finding of differentials. He finds the differentials of products — from which the differentials of quotients, powers, etc., can then be easily derived — in the following way. The product of x and y, when each is taken as reduced by half of its infinitesimal difference, becomes xy - xdy/2 - ydx/2 + dxdy/4; but if x and y are made to increase by the same amount, it becomes xy + xdy/2 + ydx/2 + dxdy/4. Now when the first product is subtracted from the second, ydx + xdy remains as a surplus and this is said to be the surplus of the increase by a whole dx and dy, for this increase is the difference between the two products; it is therefore the differential of xy. Clearly, in this procedure, the term which forms the chief difficulty, the product of the two infinitesimal differences, cancels itself out. But in spite of the name of Newton it must be said that such an operation although very elementary, is incorrect; it is not true that (x + dx/2) (y + dy/2) - (x - dx/2) (y - dy/2) = (x + dx) (y + dy) - xy. It can only have been the need to establish the all-important fluxional calculus which could bring a Newton to deceive himself with such a proof.
§ 591
Other forms which Newton employed in the derivation of differentials are bound up with concrete meanings of the elements and their powers, meanings relating to motion. About the use of the serial form which also characterises his method, it suggests itself to say that it is always possible to obtain the required degree of accuracy by adding more terms and that the omitted terms are relatively insignificant, in general, that the result is only an approximation; though here too he would have been satisfied with this ground for omission as he is in his method of solving equations of higher degree by approximation, where the higher powers arising from the substitution in the given equation of any ascertained, still inexact term, are omitted on the crude ground of their relative smallness.'
§ 592
The error into which Newton fell in solving a problem by omitting essential, higher powers, an error which gave his opponents the occasion of a triumph of their method over his, and the true origin of which has been indicated by Lagrange in his recent investigation of it demonstrates the formalism and uncertainty which still prevailed in the use of this instrument. Lagrange shows that Newton made the mistake because he omitted the term of the series containing that power on which the specific problem turned. Newton had kept to the formal, superficial principle of omitting terms on account of their relative smallness. For example, it is well known that in mechanics the terms of the series in which the function of a motion is developed are given a specific meaning, so that the first term or the first function refers to the moment of velocity, the second to the accelerating force and the third to the resistance of forces. Here, then, the terms of the series are not to be regarded merely as parts of a sum, but rather as qualitative moments Of a whole determined by the concept. In this way, the omission of the rest of the terms belonging to the spuriously infinite series acquires an altogether different meaning from omission on the ground of their relative smallness.
[Both considerations are found set simply side by side in the application by Lagrange of the theory of functions to mechanics in the chapter on rectilinear motion The space passed through, considered as a function of the time elapsed, gives the equation x = ft; this, developed as f(t + d) gives ft + df't + d2/2.f"t + , etc.
Thus the space traversed in the period of time is represented in the formula as = df't + d2f"t + d3/2.3f"'t +, etc. The motion by means of which this space has been traversed is (it is said) therefore — i.e. because the analytical development gives several, in fact infinitely, many terms — composed of various partial motions, of which the spaces corresponding to the time will be df't, d2/2f"t, d3/2.3f"'dt, etc. The first partial — notion is, in known motion, the formally uniform one with a velocity designated by f't, the second is uniformly accelerated motion derived from an accelerative force proportional to f"t. Now since the remaining terms do not refer to any simple known motion, it is not necessary to take them specially into account and we shall show that they may be abstracted from in determining the motion at the beginning of the point of time.' This is now shown, but of course only by comparing the series all of whose terms belonged to the determination of the magnitude of the space traversed in the period of time, with the equation given in art. 3 for the motion of a falling body, namely x = at + bt2 in which only these two terms occur. But this equation has itself received this form only because the explanation given to the terms produced by the analytical development is presupposed; this presupposition is that the uniformly accelerated motion is composed of a formally uniform motion continued with the velocity attained in the preceding period of time, and of an increment (the a in s = at2, i.e. the empirical coefficient) which is ascribed to the force of gravity — a distinction which has no existence or basis whatever in the nature of the thing itself, but is only the falsely physicalised expression of what issues from the assumed analytical treatment.]
The error in the Newtonian solution arose, not because terms of the series were neglected only as parts of a sum, but because the term containing the qualitative determination, which is the essential point, was ignored.
§ 593
In this example, the procedure is made to depend on the qualitative meaning. In this connection the general assertion can at once be made that the whole difficulty of the principle would be removed if the qualitative meaning of the principle were stated and the operation were made to depend on it — in place of the formalism which links the determination of the differential only to that which gives the problem its name, to the difference as such between a function and its variation after its variable magnitude has received an increment. In this sense, it is obvious that the differential of xn is completely exhausted by the first term of the series which results from the expansion of (x + dx)n . Thus the omission of the rest of the terms is not on account of their relative smallness; and so there is no assumption of an inexactitude, an error or mistake which could be compensated or rectified by another error — a point of view from which Carnot in particular justifies the ordinary method of the infinitesimal calculus. Since what is involved is not a sum but a relation, the differential is completely given by the first term; and where further terms, the differentials of higher orders, are required, their determination involves not the continuation of a series as a sum, but the repetition of one and the same relation which alone is desired and which is thus already completely given in the first term. The need for the form of a series, its summation and all that is connected with it, must then be wholly separated from the said interest of the relation.
§ 594
The explanations of the methods of infinitesimal magnitudes given by Carnot, contain a most lucid exposition of what is essential in the ideas referred to above. But in passing to the practical application itself, there enter more or less the usual ideas about the infinite smallness of the omitted terms relatively to the others. He justifies the method, not by the nature of the procedure itself, but by the fact that the results are correct, and by the advantages of a simplification and shortening of the calculus which follow the introduction of imperfect equations, as he calls them, i.e. those in which such an arithmetically incorrect omission has occurred.
§ 595
Lagrange, as is well known, reverted to Newton's original method, that of series, in order to be relieved of the difficulties inherent in the idea of the infinitely small and in the method of first and final ratios and limits. The advantages of his functional calculus as regards precision, abstraction and generality, are sufficiently recognised; we need mention only what is pertinent here, that it rests on the fundamental principle that the difference, without becoming zero, can be assumed so small that each term of the series is greater than the sum of all the following terms. This method, too, starts from the categories of increment and difference of the function, the variable magnitude of which receives the increment, thereby bringing in the troublesome series of the original function; also in the sequel the terms to be omitted are considered only as constituting a sum, while the reason for omitting them is made to consist in the relativity of their quantum. And so here, too, on the one hand, the principle of the omission is not brought back to the point of view exemplified in some applications, where (as was remarked above) terms of the series are supposed to have a specific quality significance, and terms are neglected not because of their quantitative insignificance but because they are not qualitatively significant; and then, on the other hand, the omission itself has no place in the essential point of view which, as regards the so-called differential coefficient, only becomes specifically prominent with Lagrange, in the so-called application of the calculus, as will be more fully considered in the following remark.
§ 596
The demonstrated qualitative character as such of the form of magnitude here under discussion in what is called the infinitesimal, is found most directly in the category of limit of the ratio referred to above and the carrying out of which in the calculus has been developed into a characteristic method. Lagrange criticises this method as lacking case in application and he claims that the expression limit does not present any definite idea; this second point we will take up here and examine more closely what is stated about its analytical meaning. Now the idea of limit does indeed imply the true category of the qualitatively determined relation of variable magnitudes above-mentioned; for the forms of it which occur, dx and dy, are supposed to be taken simply and solely as moments of dy/dx, and dy/dx itself must be regarded as a single indivisible symbol.
§ 597
That the mechanism of the calculus, especially in its application, thus loses the advantage it derived from the separation of the sides of the differential coefficient, this we will pass over here. Now the said limit is to be limit of a given function; it is to assign to this function a certain value determined by its mode of derivation. But with the mere category of limit we should not have advanced beyond the scope of this Remark, which is to demonstrate that the infinitely small which presents itself in the differential calculus as dx and dy, does not have merely the negative, empty meaning of a non-finite, non-given magnitude, as when one speaks of 'an infinite multitude', 'to infinity', and the like, but on the contrary has the specific meaning of the qualitative nature of what is quantitative, of a moment of a ratio as such. This category, however, merely as such, still has no relation to that which is a given function and does not itself enter into the treatment of such a function or into the use to be made of that determination; thus the idea of limit, too, confined to this its demonstrated character, would lead nowhere. But the very expression 'limit' implies that it is a limit of something, i.e. that it expresses a certain value which lies in the function of a variable magnitude; and we must examine the nature of this concrete role. It is supposed to be the limit of the ratio between the two increments by which the two variable magnitudes connected in an equation (one of which is regarded as a function of the other), are supposed to have been increased; the increase is taken here as quite undetermined and so far no use is made of the infinitely small. But the way in which this limit is found involves the same inconsistencies as are contained in the other methods. This way is as follows: if y = fx, then when y becomes y + k, fx is to change into fx + ph + qh2 + rh3 and so on; thus k = ph + qh2, etc., and k/h = p + qh + rh2, etc. Now if k and h vanish, the right-hand side of the equation also vanishes with the exception of p; now p is supposed to be the limit of the ratio of the two increments. It is clear that while h, as a quantum, is equated with 0, k/h nevertheless is not at the same time equal to 0/0 but is supposed still to remain a ratio.
§ 598
Now the idea of limit is supposed to have the advantage of avoiding the inconsistency here involved; p is, at the same time, supposed to be not the actual ratio, which would be 0/0 but only that specific value to which the ratio can infinitely approximate, i.e. can approach so near that the difference can be smaller than any given difference. The more precise meaning of approximation with respect to the terms which are supposed really to approach each other will be considered later. But that a quantitative difference, the definition of which is that it not only can, but shall be smaller than any given difference, is no longer a quantitative difference, this is self-evident, as self-evident as anything can be in mathematics; but we still have not got away from dy/dx = 0. If on the other hand dy/dx = p, i. e. is assumed to be a definite quantitative ratio as in fact it is, then conversely there is a difficulty about the presupposition which equates h with o, a presupposition which is indispensable for obtaining the equation k/h = p. But if it be granted that k/h = 0, (and when h = 0, k is in fact automatically = 0, for k, the increment of y, depends entirely on the existence of the increment h), then the question would arise, what p — which is a quite definite quantitative value — is supposed to be. To this there is at once an obvious answer, the simple, meagre answer that it is a coefficient derived in such and such a way — the first function, derived in a certain specific manner, of an original function. if we content ourselves with this — and Lagrange did, in fact, do so in practice — then the general part of the science of the differential calculus, and straightway this one particular form of it called the theory of limits would be rid of the increments and of their infinite or arbitrary smallness — spared too, the difficulty of getting rid again of all the terms of a series other than the first, or rather only the coefficient of the first, which inevitably follow on the introduction of these increments; in addition it would also be purged of those formal categories connected with them, especially of the infinite, of infinite approximation and, too, the categories, here equally empty, of continuous magnitude' which, moreover, like nisus, becoming, occasion of a variation, are deemed necessary.
§ 599
[The category of continuous or fluent magnitude enters with the consideration of the external and empirical variation of magnitudes — which are brought by an equation into the relation in which one is a function of the other; but since the scientific object of the differential calculus is a certain relation (usually expressed by the differential coefficient), the specific nature of which may equally well be called a law, the mere continuity is a heterogeneous aspect of this specific nature, and besides is in any case an abstract and here empty category seeing that nothing whatever is said about the law of continuity. Into what formal definitions one may be led in these matters can be seen from the penetrating exposition by my respected colleague, Prof. Dirksen of the fundamental determinations used in the deduction of the differential calculus, which forms an appendix to the criticism of some recent works on this science. The following definition is actually quoted: 'A continuous magnitude, a continuum, is any magnitude thought of as in a state of becoming such that this becoming takes place not by leaps but by an uninterrupted progress'. This is surely tautologically the same as what was to be defined.]
§ 600
But it would then be required to show what other meaning and value p has — apart from the meagre definition, quite adequate for the theory, that it is simply a function derived from the expansion of a binomial — i.e. what relationships it embodies and what further use can be made of them mathematically; this will be the subject of Remark 2. But first we shall proceed to discuss the confusion which the conception of approximation currently used in expositions of the calculus, has occasioned in the understanding of the true, qualitative determinateness of the relation which was the primary interest concerned.
§ 601
It has been shown that the so-called infinitesimals express the vanishing of the sides of the ratio as quanta, and that what remains is their quantitative relation solely as qualitatively determined; far from this resulting in the loss of the qualitative relation, the fact is that it is just this relation which results from the conversion of finite into infinite magnitudes. As we have seen, it is in this that the entire nature of the matter consists. Thus in the final ratio, for example, the quanta of abscissa and ordinate vanish; but the sides of this ratio essentially remain, the one an element of the ordinate, the other an element of the abscissa. This vanishing being represented as ' an infinite approximation, the previously distinguished ordinate is made to pass over into the other ordinate, and the previously distinguished abscissa into the other abscissa; but essentially this is not so, the ordinate does not pass over into the abscissa, neither does the abscissa pass into the ordinate. To continue with this example of variable magnitudes, the element of the ordinate is not to be taken as the difference of one ordinate from another ordinate, but rather as the difference or qualitative determination of magnitude relatively to the element o the abscissa; the principle of the one variable magnitude relatively to that of the other is in reciprocal relation with it. The difference, as no longer a difference of finite magnitudes, has ceased to be manifold within itself; it has collapsed into a simple intensity, into the determinateness of one qualitative moment of a ratio relatively to the other.
§ 602
This is the nature of the matter but it is obscured by the fact that what has just been called an element, for example, of the ordinate, is grasped as a difference or increment in such a way that it is only the difference between the quantum of one ordinate and the quantum of another ordinate. And so the limit here does not have the meaning of ratio; it counts only as the final value to which another magnitude of a similar kind continually approximates in such a manner that it can differ from it by as little as we please, and that the final ratio is a ratio of equality. The infinite difference is thus the fluctuation of a difference of one quantum from another quantum, and the qualitative nature according to which dx is essentially not a determination of the ratio relatively to x, but to dy, comes to be overlooked. Dx is permitted to vanish relatively to dx, but even more does dx vanish relatively to x; but this means in truth: it has a relation only to dy. In such expositions, geometricians are mainly concerned to make intelligible the approximation of a magnitude to its limit and to keep to this aspect of the difference of quantum from quantum, how it is no difference and yet still is a difference. But all the same, approximation is a category which of itself says nothing and explains nothing; dx already has approximation behind it; it is neither near nor nearer; and 'infinitely near', itself means the negation of nearness and approximation.
§ 603
Now since this implies that the increments or infinitesimals have been considered only from the side of the quantum which vanishes in them, and only as a limit, it follows that they are grasped as unrelated moments. From this would follow the inadmissible idea that it is allowed in the final ratio to equate, say abscissa and ordinate, or even sine, cosine, tangent, versed sine, and what not. This idea seems at first to prevail when the arc is treated as a tangent; for the arc, too, is certainly incommensurable with the straight line, and its element is, in the first place, of another quality than the element of the straight line. It seems even more absurd and inadmissible than the confusing of abscissa, ordinate, versed sine, cosine, etc., when quadrata rotundas, when part of an arc, even though an infinitely small part, is taken to be a part of the tangent and so treated as a straight line. However, this treatment differs essentially from the confusion we have decried; it is justified by the circumstance that in the triangle which has for its sides the element of an arc and the elements of its abscissa and ordinate, the relation is the same as if this element of the arc were the element of a straight line, of the tangent; the angles which constitute the essential relation, i.e. that which remains to these elements when abstraction is made from the finite magnitudes belonging to them, are the same. This can also be expressed as the transition of straight lines which are infinitely small, into curved lines, and their relation in their infinity as a relation of curves. Since, according to its definition, a straight line is the shortest distance between two points, its difference from the curved line is based on the determination of amount, on the smaller amount of what is differentiated in this manner, a determination, therefore, of a quantum. But this determination vanishes in the line when it is taken as an intensive magnitude, as an infinite moment, as an element, and with it, too, its difference from the curved line which rested merely on the difference of quantum. As infinite, therefore, the straight line and arc no longer retain any quantitative relation nor consequently, on the basis of the assumed definition, any qualitative difference from each other either; on the contrary, the former passes into the latter.
§ 604
Analogous, although also distinct from, the equating of heterogeneous forms is the assumption that infinitely small parts of the same whole are equal to each other; an assumption in itself indefinite and completely indifferent, but which, applied to an object heterogeneous within itself, i.e. an object whose quantitative determination is essentially non-uniform, produces the peculiar inversion contained in that proposition of higher mechanics which states that infinitely small parts of a curve are traversed in equal, infinitely small times in a uniform motion, inasmuch as this is asserted of a motion in which in equal finite, i.e. existent, parts of time, finite, i.e. existent, unequal parts of the curve are traversed, of a motion therefore which exists as non-uniform and is assumed as such. This proposition is the expression in words of what is supposed to be the significance of an analytical term obtained in the above-mentioned development of the formula relating to a motion which is non-uniform but otherwise conforms to a law. Earlier mathematicians sought to express in words and propositions and to exhibit in geometrical tables the results of the newly invented infinitesimal calculus (which moreover always had to do with concrete objects), chiefly in order to use them for theorems susceptible of the ordinary method of proof. The terms of a mathematical formula into which analytical treatment resolved the magnitude of the object, of motion, for instance, acquired there an objective significance, such as velocity, force of acceleration, and so on; in accordance with this meaning they were supposed to furnish correct propositions, physical laws; their objective connections and relations, too, were supposed to be determined in accordance with the analytical connection. A particular example is that in a uniformly accelerated motion there is supposed to exist a special velocity proportional to the times, but that to this velocity there constantly accrues an increment from the force of gravity.
§ 605
In the modern, analytical form of mechanics such propositions are put forward simply as results of the calculus, without enquiry whether by themselves and in themselves they have a real significance, i.e. one to which there is a corresponding physical existence and whether such meaning can be demonstrated. The difficulty of making intelligible the connection of such forms when they are taken in the real meaning alluded to, for example the transition from said simply uniform velocity to a uniformly accelerated velocity, is held to be completely eliminated by the analytical treatment in which such connection is a simple result of the authority now established once and for all of the operations of the calculus. It is announced as a triumph of science that by means of the calculus alone, laws are found transcending experience, that is, propositions about existence which have no existence. But in the earlier, still naive period of the infinitesimal calculus, the aim was to assign to those forms and propositions represented in geometrical diagrams a real meaning of their own and to make that meaning plausible, and to apply the forms and propositions bearing such meaning in the proof of the main propositions concerned.
§ 606
It cannot be denied that in this field much has been accepted as proof, especially with the aid of the nebulous conception of the infinitely small, for no other reason than that the result was always already known beforehand, and that the proof which was so arranged that the result did emerge, at least produced the illusion of a framework of proof, an illusion which was still preferred to mere belief or knowledge from experience. But 1 do not hesitate to regard this affectation as nothing more than mere jugglery and window-dressing, and I include in this description even Newton's proofs, especially those belonging to what has just been quoted, for which Newton has been extolled to the skies and exalted above Kepler, namely that he demonstrated mathematically what Kepler had discovered merely empirically.
§ 607
The empty scaffolding of such proofs was erected in order to prove physical laws. But mathematics is altogether incapable of proving quantitative determinations of the physical world in so far as they are laws based on the qualitative nature of the moments [of the subject matter]; and for this reason, that this science is not philosophy, does not start from the Notion, and therefore the qualitative element, in so far as it is not taken lemmatically from experience, lies outside its sphere. The desire to uphold the honour of mathematics, that all its propositions ought to be rigorously proved, has often caused it to forget its limits; thus it seemed against its honour to acknowledge simply experience as the source and sole proof of empirical propositions. Consciousness has since then developed a more instructed view of the matter; so long, however, as consciousness is not clearly aware of the distinction between what is mathematically demonstrable and what can come only from another source, between what are only terms of an analytical expansion and what are physical existences, scientific method cannot be developed into a rigorous and pure attitude in this field. Without doubt, however, the same justice will be done to that framework of Newtonian proof as was done to another baseless and artificial Newtonian structure of optical experiments and conclusions derived from them. Applied mathematics is still full of a similar concoction of experiment and reflection; but just as one part after another of Newtonian optics long since began to be ignored in practice by the science — with the inconsistency however that all the rest although in contradiction was allowed to stand — so, too, it is a fact that already some of those illusory proofs have fallen into oblivion or have been replaced by others.
Remark 2: The Purpose of the Differential Calculus - next section