Nous allons montrer dans ce texte que l’inexistence de l’infini dans la nature est dialectiquement liée à l’inexistence du zéro. Rappelons ce que signifie croire en l’existence du zéro et de l’infini.
La question des infinis est une vieille question en physique. L’un des objectifs fondamentaux de la physique moderne a consisté à chercher le moyen de s’affranchir des infinis qui apparaissaient dans les équations classiques. Cela a mené aussi bien à la mécanique quantique, à l’électrodynamique quantique, à la renormalisation qu’à un grand nombre d’autres grandes découvertes comme la relativité. Cette dernière affirme en effet qu’il existe une limite en ce qui concerne la matière et la lumière : une vitesse ne peut pas grandir indéfiniment. Elle est limitée par la vitesse de la lumière. Cependant, il n’y pas seulement une limite supérieure du mouvement mais également une limite inférieure soulignée cette fois par la physique quantique. Celle-ci affirme qu’on ne peut pas obtenir d’immobilité absolue et qu’il est donc impossible d’obtenir une précision absolue de vitesse ou de position (principe d’incertitude de la physique quantique). Quand on parle d’infinis, on parle aussi bien des infiniment petits que des infiniment grands. Et c’est lié. Ainsi, une position très finement définie (avec une erreur infiniment petite) signifie une précision infiniment grande. D’autre part, le principe d’incertitude de la physique quantique suppose que si deux quantités sont complémentaires, lorsque l’une aurait une imprécision infiniment petite, l’autre aurait une imprécision infiniment grande. Ainsi, une particule quantique qui aurait une précision infinie de vitesse aurait une position complètement inconnue. Il faudrait également une énergie infiniment grande pour avoir une précision de temps infiniment petite. L’existence des quanta signifie également qu’il n’y a pas d’infinis puisqu’en termes de matière et de lumière, on ne peut pas descendre en dessous d’un quanta de Planck. Or, les études de Planck menaient à des limites non seulement en termes d’action (le quanta est une action, c’est-à-dire un produit d’une énergie et d’un temps) mais également d’énergie, de masse, de temps, d’espace. Ces limites inférieures s’appellent le temps de Planck, la masse de Planck ou la distance de Planck. Ces limites peuvent être très petites ou très grandes mais jamais infiniment grandes ni infiniment petites, de même que la vitesse de la lumière est très grande et le quanta d’action est très petit. En fait, personne ne peut prétendre avoir jamais fait une mesure infinie. Et pourtant des phénomènes avaient des lois relativement simples apparemment comme l’interaction électromagnétique, lois mathématiques qui menaient à des infinis. Certaines de ces quantités infinies qui apparaissent dans les équations ne sont pas totalement résolues, ou ne le sont pas avec l’accord de la communauté scientifique. Donnons quelques exemples d’infinis qui apparaissent dans les équations. De nombreuses interactions donnent une formule ayant la distance au dénominateur d’une division. Cela signifie que la force ou l’énergie devient infinie dès que la distance s’annule. Or, on suppose le plus souvent que les particules occupent un espace de dimension nulle : sont ponctuelles. Soit cette image ponctuelle des particules doit être abandonnée (ce qui suppose d’autres infinis insolubles du fait des mouvements et de l’électricité), soit cela signifie qu’il n’y a pas de contact entre particules. Elles interagissent donc sans annuler leur distance. Ce qu’on appelle des chocs est seulement un relatif rapprochement. Donc, deux particules ne peuvent pas être infiniment proches, pas plus que leur énergie d’interaction ne peut être infiniment grande. En termes d’électromagnétisme, le problème s’est révélé très difficile. On a pu interpréter l’interaction entre deux particules quantiques électrisées comme interaction entre une particule et le vide couplée à l’interaction de l’autre particule et du vide. Par contre, si on envisage une particule isolée, la même formule qui a si bien marché pour l’interaction de deux particules donne un résultat infiniment grand pour l’interaction électromagnétique de la particule électrisée et du vide ! Les physiciens reconnaissent que l’on ne comprendra les interactions qu’en étant capables de comprendre le fonctionnement du vide quantique. C’est le désordre quantique du vide qui doit expliquer l’ordre émergent de la matière. Le malheur pour nous, c’est que nous ne pouvons percevoir l’espace, le temps ou les particules (éphémères) du vide qu’au travers de la matière et de la lumière, les phénomènes que nous percevons nous ou que mesurent nos expériences. En tout cas, que ce soit dans le vide, dans la matière ou dans la lumière, nous n’avons pas trouvé d’infini. Des nombres grands ou petits oui mais pas d’infiniment petit ni d’infiniment grand. Cela signifie qu’il n’est pas possible d’avoir une situation purement ponctuelle qui supposerait une précision infinie. C’est dû à la fois à la physique quantique et à la physique relativiste. Le principe d’incertitude empêche cette précision absolue, ponctuelle. L’inexistence de phénomènes définissant la simultanéité du temps, mise en évidence par la relativité d’Einstein, pose le même problème. Du coup, l’image d’un écoulement du temps ou d’un déplacement dans l’espace représentés par une droite des nombres n’est pas valide. Cette succession de points est un défi permanent à l’inexistence des infiniment petits. De même, les équations fondées sur des variables réelles, les courbes continues, les différentielles reposent tous sur les infiniment petits des mathématiques. Ils n’auraient aucune existence en physique. Est-ce pour cela qu’ils ne devraient pas fonctionner dans le calcul ? Pas nécessairement. Un calcul peut parfaitement fonctionner pour un phénomène sans décrire pour autant le mode d’action de la nature. Rappelons que l’idée d’une Terre plate fonctionne parfaitement pour un architecte ou un maçon. Ils peuvent utiliser un fil à plomb pour représenter une verticale et considérer que deux fils à plomb sont parallèles. Pourtant, tous les fils à plomb se dirigent vers le centre de gravité de la Terre et sont donc très loin d’être parallèles. L’efficacité d’une image mathématique dans un domaine des sciences n’est pas suffisante. Il faut poser le problème dans sa globalité, c’est-à-dire philosophiquement ...
suite à venir....
MOTS CLEFS :
physique quantique – relativité –
chaos déterministe – atome –
système dynamique – structures dissipatives – percolation –
non-linéarité – quanta –
boucle de rétroaction – rupture de symétrie - turbulence – mouvement brownien –
le temps -
transition de phase – criticalité - attracteur étrange – résonance –
auto-organisation – vide - révolution permanente - Zénon d’Elée - Antiquité -
Trotsky – Rosa Luxemburg –
Gould - marxisme - Marx - la révolution - l’anarchisme - le stalinisme - Socrate
Les grands progrès modernes de la physique proviennent du renoncement aux notions de zéro et d’infini. La relativité est le renoncement de l’idée de vitesse infinie de la lumière et des interactions. La quantique est le renoncement à la notion de valeur nulle de la constante de Planck. La renormalisation est le renoncement aux infiniments petits dans les interactions. « En effet, alors que la faillite de la mécanique classique – dévoilée par la théorie de la relativité – est liée à la finitude de la vitesse de la lumière (au fait que celle-ci ne soit pas égale à l’infini), on découvrit à l’orée de ce siècle des divergences entre les conclusions de la mécanique et les faits expérimentaux, divergences liées à la finitude de la constance h de Planck (au fait qu’elle ne soit pas égale à zéro). »
Einstein dans « Physique et réalité »
The successes of the differential equation paradigm were impressive and extensive. Many problems, including basic and important ones, led to equations that could be solved. A process of self-selection set in, whereby equations that could not be solved were automatically of less interest than those that could.
— Stewart, Ian ; Does god play dice ? The mathematics of chaos
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
— Hempel, Carl G.
A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history. Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.
- Tolstoï, Leon N. ; Guerre et paix
Les Grâces ne s’enfuient pas devant les intégrales et les équations différentielles.
If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.
— Bush, Vannevar
Il est un concept qui corrompt et dérègle tous les autres. Je ne parle pas du Mal, dont l’empire est circonscrit à l’éthique ; je parle de l’infini.
— Borges, Jorges Luis ; Les avatars de la tortue
dans "Science de la Logique"
The Specific Nature of the Notion of the Mathematical Infinite
The mathematical infinite has a twofold interest. On the one hand its introduction into mathematics has led to an expansion of the science and to important results ; but on the other hand it is remarkable that mathematics has not yet succeeded in justifying its use of this infinite by the Notion (Notion taken in its proper meaning). Ultimately, the justifications are based on the correctness of the results obtained with the aid of the said infinite, which correctness is proved on quite other grounds : but the justifications are not based on the clarity of the subject matter and on the operation through which the results are obtained, for it is even admitted that the operation itself is incorrect.
This alone is in itself a bad state of affairs ; such a procedure is unscientific. But it also involves the drawback that mathematics, being unaware of the nature of this its instrument because it has not mastered the metaphysics and critique of the infinite, is unable to determine the scope of its application and to secure itself against the misuse of it.
But in a philosophical respect the mathematical infinite is important because underlying it, in fact, is the notion of the genuine infinite and it is far superior to the ordinary so-called metaphysical infinite on which are based the objections to the mathematical infinite. Often, the science of mathematics can only defend itself against these objections by denying the competence of metaphysics, asserting that it has nothing to do with that science and does not have to trouble itself about metaphysical concepts so long as it operates consistently within its own sphere. Mathematics has to consider not what is true in itself but what is true in its own domain. Metaphysics, though disagreeing with the use of the mathematical infinite, cannot deny or invalidate the brilliant results obtained from it, and mathematics cannot reach clearness about the metaphysics of its own concept or, therefore, about the derivation of the modes of procedure necessitated by the use of the infinite.
If it were solely the difficulty of the Notion as such which troubled mathematics, it could ignore it without more ado since the Notion is more than merely the statement of the essential determinatenesses of a thing, that is, of the determinations of the understanding : and mathematics has seen to it that these determinatenesses are not lacking in precision ; for it is not a science which has to concern itself with the concepts of its objects and which has to generate their content by explicating the concept, even if this could be effected only by ratiocination. But mathematics, in the method of its infinite, finds a radical contradiction to that very method which is peculiar to itself and on which as a science it rests. For the infinitesimal calculus permits and requires modes of procedure which mathematics must wholly reject when operating with finite quantities, and at the same time it treats these infinite quantities as if they were finite and insists on applying to the former the same modes of operation which are valid for the latter ; it is a cardinal feature in the development of this science that it has succeeded in applying to transcendental determinations and their treatment the form of ordinary calculation.
Mathematics shows that, in spite of the clash between its modes of procedure, results obtained by the use of the infinite completely agree with those found by the strictly mathematical, namely, geometrical and analytical method. But in the first place, this does not apply to every result and the introduction of the infinite is not for the sole purpose of shortening the ordinary method but in order to obtain results which this method is unable to secure. Secondly, success does not by itself justify the mode of procedure. This procedure of the infinitesimal calculus shows itself burdened with a seeming inexactitude, namely, having increased finite magnitudes by an infinitely small quantity, this quantity is in the subsequent operation in part retained and in part ignored. The peculiarity of this procedure is that in spite of the admitted inexactitude, a result is obtained which is not merely fairly close and such that the difference can be ignored, but is perfectly exact. In the operation itself, however, which precedes the result, one cannot dispense with the conception that a quantity is not equal to nothing, yet is so inconsiderable that it can be left out of account. However, what is to be understood by mathematical determinateness altogether rules out any distinction of a greater or lesser degree of exactitude, just as in philosophy there can be no question of greater or less probability but solely of Truth. Even if the method and use of the infinite is justified by the result, it is nevertheless not so superfluous to demand its justification as it seems in the case of the nose to ask for a proof of the right to use it. For mathematical knowledge is scientific knowledge, so that the proof is essential ; and even with respect to results it is a fact that a rigorous mathematical method does not stamp all of them with the mark of success, which in any case is only external.
It is worth while considering more closely the mathematical concept of the infinite together with the most noteworthy of the attempts aimed at justifying its use and eliminating the difficulty with which the method feels itself burdened. The consideration of these justifications and characteristics of the mathematical infinite which I shall undertake at some length in this Remark will at the same time throw the best light on the nature of the true Notion itself and show how this latter was vaguely present as a basis for those procedures.
The usual definition of the mathematical infinite is that it is a magnitude than which there is no greater (when it is defined as the infinitely great) or no smaller (when it is defined as the infinitely small), or in the former case is greater than, in the latter case smaller than, any given magnitude. It is true that in this definition the true Notion is not expressed but only, as already remarked, the same contradiction which is present in the infinite progress ; but let us see what is implicitly contained in it. In mathematics a magnitude is defined as that which can be increased or diminished ; in general, as an indifferent limit. Now since the infinitely great or small is that which cannot be increased or diminished, it is in fact no longer a quantum as such.
This is a necessary and direct consequence. But it is just the reflection that quantum (and in this remark quantum as such, as we find it, I call finite quantum) is sublated, which is usually not made, and which creates the difficulty for ordinary thinking ; for quantum in so far as it is infinite is required to be thought as sublated, as something which is not a quantum but yet retains its quantitative character.
To quote Kant’s opinion of the said definition which he finds does not accord with what is understood by an infinite whole : ’According to the usual concept, a magnitude is infinite beyond which there can be no greater (i.e. greater than the amount contained therein of a given unit) ; but there can be no greatest amount because one or more units can always be added to it. But our concept of an infinite whole does not represent how great it is and it is not therefore the concept of a maximum (or a minimum) ; this concept rather expresses only the relation of the whole to an arbitrarily assumed unit, with respect to which the relation is greater than any number. According as this assumed unit is greater or smaller, the infinite would be greater or smaller. The infinity, however, since it consists solely in the relation to this given unit, would always remain the same, although of course the absolute magnitude of the whole would not be known through it.’
Kant objects to infinite wholes being regarded as a maximum, as a completed amount of a given unit. The maximum or minimum as such still appears as a quantum, an amount. Such a conception cannot avert the conclusion, adduced by Kant, which leads to a greater or lesser infinite. And in general, so long as the infinite is represented as a quantum, the distinction of greater or less still applies to it. This criticism does not however apply to the Notion of the genuine mathematical infinite, of the infinite difference, for this is no longer a finite quantum.
Kant’s concept of infinite, on the other hand, which he calls truly transcendental is ’that the successive synthesis of the unit in the measurement of a quantum can never be completed’. A quantum as such is presupposed as given ; by synthesising the unit this is supposed to be converted into an amount, into a definite assignable quantum ; but this synthesis, it is said, can never be completed. It is evident from this that we have here nothing but an expression of the progress to infinity, only represented transcendentally, i.e. properly speaking, subjectively and psychologically. True, in itself the quantum is supposed to be completed ; but transcendentally, namely in the subject which gives it a relation to a unit, the quantum comes to be determined only as incomplete and as simply burdened with a beyond. Here, therefore, there is no advance beyond the contradiction contained in quantity ; but the contradiction is distributed between the object and the subject, limitedness being ascribed to the former, and to the latter the progress to infinity, in its spurious sense, beyond every assigned determinateness.
On the other hand, it was said above that the character of the mathematical infinite and the way it is used in higher analysis corresponds to the Notion of the genuine infinite ; the comparison of the two determinations will now be developed in detail. In the first place, as regards the true infinite quantum, it was characterised as in its own self infinite ; it is such because, as we have seen, the finite quantum or quantum as such and its beyond, the spurious infinite, are equally sublated. Thus the sublated quantum has returned into a simple unity and self-relation ; but not merely like the extensive quantum which, in passing into intensive quantum, has its determinateness only in itself [or implicitly] in an external plurality, towards which, however, it is indifferent and from which it is supposed to be distinct.
The infinite quantum, on the contrary, contains within itself first externality and secondly the negation of it ; it is thus no longer any finite quantum, not a quantitative determinateness which would have a determinate being as quantum ; it is simple, and therefore only a moment. It is a quantitative determinateness in qualitative form ; its infinity consists in its being a qualitative determinateness. As such moment, it is in essential unity with its other, and is only as determined by this its other, i.e. it has meaning solely with reference to that which stands in relation to it. Apart from this relation it is a nullity — simply because quantum as such is indifferent to the relation, yet in the relation is supposed to be an immediate, inert determination. As only a moment, it is, in the relation, not an independent, indifferent something ; the quantum in its infinity is a being-for-self, for it is at the same time a quantitative determinateness only in the form of a being-for-one.
The Notion of the infinite as abstractly expounded here will show itself to be the basis of the mathematical infinite and the Notion itself will become clearer if we consider the various stages in the expression of a quantum as moment of a ratio, from the lowest where it is still also a quantum as such, to the higher where it acquires the meaning and the expression of a properly infinite magnitude.
Let us then first take quantum in the relation where it is a fractional number. Such fraction, 2/7 for example, is not a quantum like 1, 2, 3, etc. ; although it is an ordinary finite number it is not an immediate one like the whole numbers but, as a fraction, is directly determined by two other numbers which are related to each other as amount and unit, the unit itself being a specific amount. However, if we abstract from this more precise determination of them and consider them solely as quanta in the qualitative relation in which they are here, then 2 and 7 are indifferent quanta ; but since they appear here only as moments, the one of the other, and consequently of a third (of the quantum which is called the exponent), they directly count no longer simply as 2 and 7 but only according to the specific relationship in which they stand to each other. In their place, therefore, we can just as well put 4 and 14, or 6 and 21, and so on to infinity. With this, then, they begin to have a qualitative character. If 2 and 7 counted as mere quanta, then 2 is just 2 and nothing more, and 7 is simply 7 ; 4, 14, 6, 21 etc., are completely different from them and, as only immediate quanta, cannot be substituted for them. But in so far as 2 and 7 are not to be taken as such immediate quanta their indifferent limit is sublated ; on this side therefore they contain the moment of infinity, since not only are they no longer merely 2 and 7, but their quantitative determinateness remains — but as one which is in itself qualitative, namely in accordance with their significance as moments in the ratio. Their place can be taken by infinitely many others without the value of the fraction being altered, by virtue of the determinateness possessed by the ratio.
However, the representation of infinity by a fractional number is still imperfect because the two sides of the fraction, 2 and 7, can be taken out of the relation and are ordinary, indifferent quanta ; their connection as moments of the ratio is an external circumstance which does not directly concern them. Their relation, too, is itself an ordinary quantum, the exponent of the ratio.
The letters with which general arithmetic operates, the next universality into which numbers are raised, do not possess the property of having a specific numerical value ; they are only general symbols and indeterminate possibilities of any specific value. The fraction a/b seems, therefore, to be a more suitable expression of the infinite, since a and b, taken out of their relation to each other, remain undetermined, and taken separately, too, have no special peculiar value. However, although these letters are posited as indeterminate magnitudes their meaning is to be some finite quantum. Therefore, though they are the general representation of number, it is only of a determinate number, and the fact that they are in a ratio is likewise an inessential circumstance and they retain their value outside it.
If we consider more closely what is present in the ratio we find that it contains the following two determinations : first it is a quantum, secondly, however, this quantum is not immediate but contains qualitative opposition ; at the same time it remains therein a determinate, indifferent quantum by virtue of the fact that it returns into itself from its otherness, from the opposition, and so also is infinite. These two determinations are represented in the following familiar form developed in their difference from each other.
The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series ; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as an aggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here ; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series ; but they can all be so expressed in a numerical system based on another unit.
Now in the infinite series, which is supposed to represent the fraction as an amount, the aspect of the fraction as a ratio has vanished and with it there has vanished too the aspect which, as we have already shown, makes the fraction in its own self infinite. But this infinity has entered in another way ; the series, namely, is itself infinite.
Now the nature of this infinity of the series is self-evident ; it is the spurious infinity of the progression. The series contains and exhibits the contradiction of representing that which is a relation possessing a qualitative nature, as devoid of relation, as a mere quantum, as an amount. The consequence of this is that the amount which is expressed in the series always lacks something, so that in order to reach the required determinateness, we must always go further than the terms already posited. The law of the progression is known, it is implicit in the determination of the quantum contained in the fraction and in the nature of the form in which it is supposed to be expressed. By continuing the series the amount can of course be made as accurate as required ; but representation by means of the series continues to remain only an ought-to-be ; it is burdened with a beyond which cannot be sublated, because to express as an amount that which rests on a qualitative determinateness is a lasting contradiction.
In this infinite series, this inexactitude is actually present, whereas in the genuine mathematical infinite there is only an appearance of inexactitude. These two kinds of mathematical infinite are as little to be confounded as are the two kinds of philosophical infinite. In representing the genuine mathematical infinite, the form of series was used originally and it has recently again been invoked ; but this form is not necessary for it. On the contrary, the infinite of the infinite series is essentially different from the genuine infinite as the sequel will show. Indeed the form of infinite series is even inferior to the fractional expression.
For the infinite series contains the spurious infinity, because what the series is meant to express remains an ought-to-be and what it does express is burdened with a beyond which does not vanish and differs from what was meant to be expressed. It is infinite not because of the terms actually expressed but because they are incomplete, because the other which essentially belongs to them is beyond them ; what is really present in the series, no matter how many terms there may be, is only something finite, in the proper meaning of that word, posited as finite, i.e., as something which is not what it ought to be. But on the other hand, what is called the finite expression or the sum of such a series lacks nothing ; it contains that complete value which the series only seeks ; the beyond is recalled from its flight ; what it is and what it ought to be are not separate but the same.
What distinguishes these two is more precisely this, that in the infinite series the negative is outside its terms which are present only qua parts of the amount. On the other hand, in the finite expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of the ratio and this is an accomplished return-into-self, a self-related unity as a negation of the negation (both sides of the ratio are only moments), and consequently has within it the determination of infinity. Thus the usually so-called sum, the 2/7 or 1/(1 - a) is in fact a ratio ; and this so-called finite expression is the truly infinite expression. The infinite series, on the other hand, is in truth a sum ; its purpose is to represent in the form of a sum what is in itself a ratio, and the existing terms of the series are not terms of a ratio but of an aggregate. Furthermore, the series is in fact the finite expression ; for it is the incomplete aggregate and remains essentially deficient. According to what is really present in it, it is a specific quantum, but at the same time it is less than what it ought to be ; and then, too, what it lacks is itself a specific quantum ; this missing part is in fact that which is called infinite in the series, from the merely formal point of view that it is something lacking, a non-being ; with respect to its content it is a finite quantum. Only what is actually present in the series, plus what is lacking, together constitute the amount of the fraction, the specific quantum which the series also ought to be but is not capable of being. The word infinite, even as used in infinite series, is commonly fancied to be something lofty and exalted ; this is a kind of superstition, the superstition of the understanding ; we have seen how, on the contrary, it indicates only a deficiency.
We may further remark that the existence of infinite series which cannot be summed is an external and contingent circumstance with respect to the form of series as such. They contain a higher kind of infinity than do those which can be summed, namely an incommensurability, or the impossibility of representing the quantitative ratio contained in them as a quantum, even in the form of a fraction ; but the form of series as such which they have contains the same determination of spurious infinity that is present in the series capable of summation.
The terminological inversion we have just noticed in connection with the fraction and its expression as a series, also occurs when the mathematical infinite — not the one just named but the genuine infinite — is called the relative infinite, while the ordinary metaphysical — by which is understood the abstract, spurious infinite is called absolute. But in point of fact it is this metaphysical infinite which is merely relative, because the negation which it expresses is opposed to a limit only in such a manner that this limit persists outside it and is not sublated by it ; the mathematical infinite, on the contrary, has within itself truly sublated the finite limit because the beyond of the latter is united with it.
It is primarily in this sense, in which it has been demonstrated that the so-called sum or finite expression of an infinite series is rather to be regarded as the infinite expression, that Spinoza opposes the concept of true infinity to that of the spurious and illustrates it by examples. It will shed most light on his concept if I follow up this exposition with what he says on the subject.
He starts by defining the infinite as the absolute affirmation of any kind of natural existence, the finite on the contrary as a determinateness, as a negation. That is to say, the absolute affirmation of an existence is to be taken as its relation to itself, its not being dependent on an other ; the finite, on the other hand, is negation, a ceasing-to-be in the form of a relation to an other which begins outside it. Now the absolute affirmation of an existence does not, it is true, exhaust the notion of infinity ; this implies that infinity is an affirmation, not as immediate, but only as restored by the reflection of the other into itself, or as negation of the negative. But with Spinoza, substance and its absolute unity has the form of an inert unity, i.e. of a unity which is not self-mediated, of a fixity or rigidity in which the Notion of the negative unity of the self, i.e. subjectivity, is still lacking.
The mathematical example with which he illustrates the true infinite is a space between two unequal circles which are not concentric, one of which lies inside the other without touching it. It seems that he thought highly of this figure and of the concept which it was used to illustrate, making it the motto of his Ethics. ’Mathematicians conclude’, he says, ’that the inequalities possible in such a space are infinite, not from the infinite amount of parts, for its size is fixed and limited and 1 can assume larger and smaller such spaces, but because the nature of the fact surpasses every determinateness.’ It is evident that Spinoza rejects that conception of the infinite which represents it as an amount or as a series which is not completed, and he points out that here, in the space of his example, the infinite is not beyond, but actually present and complete ; this space is bounded, but it is infinite ’because the nature of the fact surpasses every determinateness’, because the determination of magnitude contained in it cannot at the same time be represented as a quantum, or in Kant’s words already quoted, the synthesis cannot be completed to form a (discrete) quantum. How in general the opposition of continuous and discrete quantum leads to the infinite, will be shown in detail in a later Remark. Spinoza calls the infinite of a series the infinite of the imagination ; on the other hand, the infinite as self-relation he calls the infinite of thought, or infinitum actu. It is, namely, actu, actually infinite because it is complete and present within itself. Thus the series 0.285714 ... or 1 + a + a2 + a3 ... is the infinite merely of imagination or supposition ; for it has no actuality, it definitely lacks something ; on the other hand 2/7 or 1/(1 - a) is actually not only what the series is in its developed terms, but is, in addition, what the series lacks, what it only ought to be. The 2/7 or 1/(1 - a) is equally a finite magnitude like Spinoza’s space enclosed between two circles, with its inequalities, and can like this space be made larger or smaller. But this does not involve the absurdity of a larger or smaller infinite ; for this quantum of the whole does not concern the relation of its moments, the nature of the fact, i.e. the qualitative determination of magnitude ; what is actually present in the infinite series is equally a finite quantum, but it is also still deficient. Imagination on the contrary stops short at quantum as such and does not reflect on the qualitative relation which constitutes the ground of the existing incommensurability.
The incommensurability which lies in Spinoza’s example embraces in general the functions of curved lines and more precisely, leads to the infinite which mathematics has introduced with such functions, in general, with the functions of variable magnitudes. This infinite is the genuine mathematical qualitative infinite which Spinoza also had in mind. We shall now consider this determination here in detail.
First of all, as regards the category of variability which is accorded such importance and which embraces the magnitudes related in these functions, it is to be noted that these magnitudes are not supposed to be variable in the way that the two numbers 2 and 7 are in the fraction 2/7 : their place can equally well be taken by 4 and 14, 6 and 21, and by other numbers ad infinitum without altering the value of the fraction ; and still more in a/b, can a and b be replaced by any arbitrary number without altering what a/b is intended to express. Now in the sense that in the place, too, of x and y of a function, there can be put an infinite, i.e. inexhaustible, multitude of numbers, a and b are just as much variable magnitudes as the said x and y. The expression ’variable magnitudes’ is therefore very vague and ill-chosen for those determinations of magnitude whose interest and manner of treatment lie in something quite distinct from their mere variability.
In order to make clear wherein lies the true character of those moments of a function with which higher analysis is concerned, we must once more run through the stages to which we have already drawn attention. In 2/7 or a/b, 2 and 7 are each independent determinate quanta and the relation is not essential to them ; a and b likewise are intended to represent quanta which remain what they are even outside the relation. And further, 2/7 and a/b are each a fixed quantum, a quotient ; the ratio constitutes an amount, the unit of which is expressed by the denominator and the amount of these units by the numerator, or conversely ; even if 4 and 14, and so on, are substituted for 2 and 7, the ratio, also as a quantum, remains the same. But now in the function y2/x = p, for example, this is essentially changed ; here, it is true that x and y can stand for definite quanta, but only x and y2 have a determinate quotient, not x and y. Hence not only are these sides of the ratio x and y, not any determinate quanta, but, secondly, their ratio is not a fixed quantum (nor is such a quantum meant as in the case of a and b), not a fixed quotient, but this quotient is, as a quantum, absolutely variable. But this is solely because x has a relation, not to y, but to the square of y. The relation of a magnitude to a power is not a quantum, but essentially a qualitative relation ; the power-relation is the feature which is to be regarded as the fundamental determination. But in the function of the straight line y = ax, a is an ordinary fraction and quotient ; consequently this function is only formally a function of variable magnitudes, or x and y here are what a and b are in a/b that is, they are not in that determination in which the differential and integral calculus considers them. On account of the special nature of the variable magnitudes in this mode of consideration, it would have been fitting to have introduced both a special name for them and other symbols than those generally used for unknown quantities in any finite equation, determinate or indeterminate ; for there is an essential difference between those magnitudes and such quanta which are merely unknown, but are in themselves completely determined or are a definite range of determinate quanta. It is, too, only because of a lack of awareness of what constitutes the peculiar interest of higher analysis and of what has led to the need for and invention of the differential calculus, that functions of the first degree and the equation of the straight line are themselves included in the treatment of this calculus ; such formalism originates partly, too, in the mistake of imagining that the intrinsically correct demand for the generalisation of a method has been fulfilled when the specific determinateness on which the need for the calculus is based is omitted, as if in this domain we were concerned only with variable magnitudes. A great deal of formalism would, indeed, have been avoided if it had been perceived that the calculus is concerned not with variable magnitudes as such but with relations of powers.
But there is still another stage where the peculiar character of the mathematical infinite becomes prominent. In an equation in which x and y are determined primarily by a power-relation, x and y as such are still supposed to signify quanta ; now this significance is altogether and completely lost in the so-called infinitesimal differences. Dx, dy, are no longer quanta, nor are they supposed to signify quanta ; it is solely in their relation to each other that they have any meaning, a meaning merely as moments. They are no longer something (something taken as a quantum), not finite differences ; but neither are they nothing ; not empty nullities. Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as determinations of the differential coefficient dx/dy.
In this concept of the infinite, the quantum is genuinely completed into a qualitative reality ; it is posited as actually infinite ; it is sublated not merely as this or that quantum but as quantum generally. But the quantitative determinateness remains as element of the principle of the quanta, or, as has also been said, the quanta remain in their first concept.
It is this concept which has been the target for all the attacks made on the fundamental determination of the mathematics of this infinite, i.e. of the differential and integral calculus. Failure to recognise it was the result of incorrect ideas on the part of mathematicians themselves ; but it is the inability to justify the object as Notion which is mainly responsible for these attacks. But mathematics, as we remarked above, cannot evade the Notion here ; for, as mathematics of the infinite, it does not confine itself to the finite determinateness of its objects (as in ordinary mathematics, which considers and relates space and number and their determinations only according to their finitude) ; on the contrary, when it treats a determination taken from ordinary mathematics, it converts it into an identity with its opposite, e.g. converting a curved line into a straight one, the circle into a polygon, etc. Consequently, the operations which it allows itself to perform in the differential and integral calculus are in complete contradiction with the nature of merely finite determinations and their relations and would therefore have to be justified solely by the Notion.
Although the mathematics of the infinite maintained that these quantitative determinations are vanishing magnitudes, i.e. magnitudes which are no longer any particular quantum and yet are not nothing but are still a determinateness relatively to an other, it seemed perfectly clear that such an intermediate state, as it was called, between being and nothing does not exist. What we are to think of this objection and the so-called intermediate state, has already been indicated above in Remark 4 to the category of becoming. The unity of being and nothing is, of course, not a state ; a state would be a determination of being and nothing into which these moments might be supposed to have lapsed only by accident, as it were, into a diseased condition externally induced through erroneous thinking ; on the contrary, this mean and unity, the vanishing or equally the becoming is alone their truth.
Further, it has been said that what is infinite is not comparable as something greater or smaller ; therefore there cannot be a relation between infinites according to orders or dignities of the infinite, although in the science of infinitesimals these distinctions do occur. Underlying this objection already mentioned is always the idea that here we are supposed to be dealing with quanta which are compared as quanta, that determinations which are no longer quanta no longer have any relationship to each other. But the truth is rather that that which has being solely in the ratio is not a quantum ; the nature of quantum is such that it is supposed to have a completely indifferent existence apart from its ratio, and its difference from another quantum is supposed not to concern its own determination ; on the other hand the qualitative is what it is only in its distinction from an other. The said infinite magnitudes, therefore, are not merely comparable, but they exist only as moments of comparison, i.e. of the ratio.
I will adduce the most important definitions of this infinite which have been given in mathematics. From these it will be clear that the thought underlying them accords with the Notion developed here, but that the originators of the definitions did not establish the thought as Notion and found it necessary in the application to resort again to expedients which conflict with their better cause.
The thought cannot be more correctly determined than in the way Newton has stated it. I eliminate here those determinations which belong to the idea of motion and velocity (from which, mainly, he took the name of fluxions) because in them the thought does not appear in its proper abstraction but as concrete and mixed with non-essential forms. Newton explains that he understands by these fluxions not indivisibles (a form which was used by earlier mathematicians, Cavalieri and others and which involves the concept of an intrinsically determinate quantum), but vanishing divisibles ; also not sums and ratios of determinate parts but the limits (limites) of sums and ratios. It may be objected that vanishing magnitudes do not have a final ratio, because the ratio before it vanishes is not final, and when it has vanished is no longer a ratio. But by the ratio of vanishing magnitudes is to be understood not the ratio before which and after which they vanish, but with which they vanish. (quacum evanescunt). Similarly, the first ratio of nascent magnitudes is that with which they become.
Newton did what the scientific method of his time demanded, he only explained what was to be understood by an expression ; but that such and such is to be understood by it is, properly speaking, a subjective presumption, or a historical demand, without any indication that such a concept is in itself absolutely necessary or that there is truth in it. However, what has been quoted shows that the concept put forward by Newton corresponds to the way in which infinite quantity resulted from the reflection of quantum into itself in the exposition above. By magnitudes is understood magnitudes in their vanishing, i.e. which are no longer quanta ; also, not ratios of determinate parts, but the limits of the ratio. The meaning is, therefore, that with the vanishing of the quanta individually, the sides of the ratio, there also vanishes the ratio itself in so far as it is a quantum ; the limit of the quantitative ratio is that in which it both is and is not, or, more precisely, in which the quantum has vanished, with the result that the ratio and its sides are preserved, the former only as a qualitative relation of quantity and the latter similarly as qualitative moments of quantity. Newton goes on to add that from the fact that there are final ratios of vanishing magnitudes, it must not be inferred that there are final magnitudes, indivisibles. For this would mean a leap again from the abstract ratio to its sides as supposedly having an independent value of their own as indivisibles outside their relation, as something which would be a one, something devoid of any relation at all.
To prevent such a misunderstanding, he again points out that final ratios are not ratios of final magnitudes, but are limits to which the ratios of the magnitudes decreasing without limit are nearer than any given, i.e. finite, difference ; the ratios, however, do not exceed these limits, for if they did they would become nullities. In other words, final magnitudes could have been taken to mean, as already said, indivisibles or ones. But the definition of the final ratio excludes the conception both of the indifferent one which is devoid of any relation, and of the finite quantum. If the required determination had been developed into the Notion of a quantitative determination which is purely a moment of the ratio, there would have been no need for the decreasing without limit into which Newton converts the quantum and which only expresses the progress to infinity, or for the determination of divisibility which no longer has any immediate meaning here.
As regards the preservation of the ratio in the vanishing of the quanta, there is found elsewhere, as in Carnot, the expression that by virtue of the law of continuity, the vanishing magnitudes still retain the ratio from which they come, before they vanish.
This conception expresses the true nature of the matter, if the continuity of the quantum is not understood to be the continuity which it has in the infinite progress where the quantum is continued in its vanishing in such a manner that in its beyond there arises only a finite quantum again, only a fresh term of the series ; but a continuous progress is always imagined as one in which values are passed through, values which are still finite quanta. On the other hand, where the transition is made into the true infinite it is the ratio that is continuous ; so continuous is it, so completely is it preserved, that the transition may be said to consist solely in throwing into relief the pure ratio and causing the non-relational determination — i.e. that a quantum which is a side of the ratio is still a quantum outside this relation — to vanish. This purification of the quantitative ratio is thus analogous to grasping an empirical reality in terms of its Notion. The empirical reality is thereby raised above itself in such a way that its Notion contains the same characteristic features as it has itself, but these are grasped in their essentiality and are taken into the unity of the Notion in which they have lost their indifferent, Notion — less existence.
The other form of Newton’s exposition of the magnitudes in question is equally interesting, namely, as generative magnitudes or principles. A generated magnitude (genita) is a product or quotient, such as a root, rectangle, square, also the sides of rectangles and squares — in general, a finite magnitude. ’Such a magnitude being considered as variable, increasing or decreasing in ceaseless motion and flux, he gives its momentary increments or decrements the name of moments. But these are not to be taken for particles of a definite magnitude (particulae finitae) : such would not themselves be moments but magnitudes generated from moments. Rather are they to be understood as the nascent principles or beginnings of finite magnitudes.’ Here the quantum is distinguished from itself : as a product or a real being [Daseiendes], and in its becoming (or as nascent), in its beginning and principle, that is to say, in its Notion, or, what is here the same thing, in its qualitative determination : in the latter the quantitative differences, the infinite increments or decrements, are only moments ; only that which has becoming at its back has passed over into the indifference of determinate being and into externality, i.e. is quantum. But if on the one hand the philosophy of the true Notion must acknowledge these determinations of the infinite with respect to increments or decrements, on the other hand it must be observed that the very forms of increments etc. fall within the category of immediate quantum and of the continuous progress to which we have referred ; in fact the conceptions of increment, growth or increase of x by dx or i, and so on, are to be regarded as the fundamental vice in these methods — the permanent obstacle to disengaging the determination of the qualitative moment of quantity in its purity from the conception of the ordinary quantum.
The conception of infinitesimals which is implicit, too, in the increment or decrement itself, is much inferior to the above determinations. The nature of these magnitudes is supposed to be such that they may be neglected, not only in comparison with finite magnitudes, but also their higher orders in comparison with their lower, and even the products of several in comparison with a single one. With Leibniz, this demand to neglect is more strikingly prominent than with previous inventors of methods relating to these infinitesimals in which this call to neglect also occurs. It is chiefly this call to neglect which, along with a gain in facility, has given this calculus the appearance of inexactitude and express incorrectness in its method of procedure. Wolf has tried to make this neglect intelligible in his own way of popularising things, i.e. by polluting the pure Notion and setting in its place incorrect sensuous conceptions. For example, he compares the neglect of infinitesimals of higher orders relatively to lower with the procedure of a surveyor who, in measuring the height of a mountain is no less accurate if meanwhile the wind has blown away a grain of sand from the top ; or with the neglect of the height of houses or towers when calculating lunar eclipses.
Even if ordinary common sense in fairness allows such inexactitude, all geometricians reject this conception. It is quite obvious that in the science of mathematics there cannot be any question of such empirical accuracy ; mathematical measuring by operations of the calculus or by geometrical constructions and proofs is altogether different from land-surveying, from the measuring of empirical lines, figures etc. Besides, by comparing the result obtained by a strictly geometrical method with that obtained by the method of infinite differences, analysts demonstrate that the one is the same as the other and that there is absolutely no question of a greater or lesser degree of exactness. And it is self-evident that an absolutely exact result could not emerge from an inexact method. Yet on the other hand again, the method itself cannot do without this omission of what is regarded as insignificant, despite its protestations against the way this omission is justified. And this is the difficulty which engages the efforts of the analysts to make intelligible and to remove the inherent inconsistency.
It is especially Euler’s conception of the matter which must be cited here. He adopts the general Newtonian definition and insists that the differential calculus considers the ratios of the increments of a magnitude, but that the infinite difference as such is to be considered as wholly nil. How this is to be understood is clear from the foregoing ; the infinite difference is a nil only of quantum, not a qualitative nil, but as a nil of quantum it is a pure moment of the ratio only. It is not a quantitative difference ; but for that reason it is, on the one hand, altogether wrong to speak of those moments which are called infinitesimals, also as increments or decrements and as differences. This description implies that something is added to or subtracted from the initially given finite magnitude, that a subtraction or addition, an arithmetical, external operation takes place. But it is to be noticed that the transition of the function of the variable magnitude into its differential is of a quite different nature ; as we have made clear, it is to be considered as a reduction of the finite function to the qualitative relation of its quantitative determinations. On the other hand, the error becomes obvious when it is said that the increments by themselves are zeros, that only their ratios are considered ; for a zero no longer has any determinateness at all. This conception then, does get as far as the negative of the quantum and gives definite expression to it, but at the same time it does not grasp this negative in its positive significance of qualitative determinations of quantity which, if they were torn out of the ratio and regarded as quanta, would be only zeros.
The opinion of Lagrange on the idea of limits or final ratios is that although one can well imagine the ratio of two magnitudes so long as they remain finite, this ratio does not present any clear and definite concept to the intellect as soon as its terms become simultaneously zero. And the understanding must, indeed, transcend this merely negative side on which the terms of the ratio are quantitatively zero, and must grasp them positively, as qualitative moments. But we cannot regard, as satisfactory Euler’s further remarks with regard to this conception of his in which he tries to show that two so-called infinitesimals which are supposed to be nothing else but zeros, nevertheless stand in a relation to each other, for which reason they are denoted by symbols other than zero. He tries to base this on the difference between the arithmetical and geometrical ratio : in the former, we have an eye to the difference, in the latter, to the quotient, so that although in the former there is no difference between two zeros, this is not so in the geometrical ratio ; if 2 : 1 = 0 : 0 then from the nature of proportion it follows that, because the first term is twice as great as the second, the third is also twice as great as the fourth ; thus according to proportion, 0 : 0 is to be taken as the ratio of 2 : 1. Even in common arithmetic n. 0 = 0 and therefore n : 1 = 0 : 0. But it is just because 2 : 1 or n : 1 is a relation of quanta that there cannot be any corresponding ratio or expression of 0 : 0.
I refrain from citing any further instances since those already considered show clearly enough that the genuine Notion of the infinite is, in fact, implied in them, but that the specific nature of that Notion has not been brought to notice and grasped. Consequently, in the actual application of the method of infinitesimals, the genuine Notion of the infinite cannot exercise any influence ; on the contrary, there is a return of the finite determinateness of quantity and the operation cannot dispense with the conception of a quantum which is merely relatively small. The calculus makes it necessary to subject the so-called infinitesimals to ordinary arithmetical operations of addition and so on, which are based on the nature of finite magnitudes, and therefore to regard them momentarily as finite magnitudes and to treat them as such. It is for the calculus to justify its procedure in which it first brings them down into this sphere and treats them as increments or differences, and then neglects them as quanta after it had just applied forms and laws of finite magnitudes to them.
I will proceed to cite the main features of the attempts of the geometricians to remove these difficulties.
The older analysts had little scruples in the matter, but the moderns directed their efforts mainly towards bringing the differential calculus back to the evidence of a strictly geometrical method and in it to attain to the rigour of the proofs of the ancients (Lagrange’s expressions) in mathematics. But since the principle of infinitesimal analysis is of a higher nature than the principle of the mathematics of finite magnitudes, that kind of evidence had perforce to be dispensed with, just as philosophy, too, cannot lay claim to that obviousness which belongs to the natural sciences, e.g. natural history — and just as eating and drinking are reckoned a more intelligible business than thinking and understanding. Accordingly, we shall deal only with the efforts to attain to the rigour of proof of the ancients.
Some have attempted to dispense altogether with the concept of the infinite, and without it to achieve what seemed to be bound up with its use. Lagrange speaks, e.g., of the method devised by Landen, saying that it is purely analytical and does not employ infinitesimal differences, but starts with different values of variable magnitudes and subsequently equates them. He also gives it as his opinion that in this method, the differential calculus loses its own peculiar advantages, namely simplicity of method and facility of operation. This is, indeed, a procedure which in some measure corresponds to the starting-point of Descartes’ tangential method of which detailed mention will be made later. This much, we may remark here, is generally evident, that the general procedure in which different values of variable magnitudes are assumed and subsequently equated, belongs to another department of mathematical treatment than that to which the method of the differential calculus itself belongs ; and that the peculiar nature of the simple relation (to be considered in detail further on) to which its actual, concrete determination reduces, namely, of the derived function to the original, is not brought into prominence.
The earlier of the moderns, Fermat, Barrow, and others for example, who at first used infinitesimals in that application which was subsequently developed into the differential and integral calculus, and then Leibniz, too, and those following him including Euler, always frankly believed that they were entitled to omit the products of infinitesimal differences and their higher powers, solely on the ground that they vanish relatively to the lower order. This is for them the sole basis of the fundamental principle, namely the determination of that which is the differential of a product or a power, for the entire theoretical teaching reduces to this. The rest is partly the mechanism of development and partly application, in which however as we shall later on see, the more important, or rather the sole, interest is to be found. With respect to the present topic, we need only mention here what is elementary, that on the same ground of insignificance, the cardinal principle adopted in relation to curves is that the elements of the curves, namely the increments of abscissa and ordinate, have the relation to each other of subtangent and ordinate ; for the purpose of obtaining similar triangles, the arc which forms the third side of a triangle to the two increments of the characteristic triangle (as it rightly used to be called), is regarded as a straight line, as part of the tangent and one of the increments therefore as reaching to the tangent. By these assumptions those determinations are, on the one hand, raised above the nature of finite magnitudes, but on the other hand, a method which is valid only for finite magnitudes and which does not permit the omission of anything on the ground of insignificance, is applied to moments now called infinitesimal. With such a mode of procedure, the difficulty which encumbers the method remains in all its starkness.
We must mention here a remarkable procedure of Newton the invention of an ingenious device to remove the arithmetically incorrect omission of the products of infinitesimal differences or higher orders of them in the finding of differentials. He finds the differentials of products — from which the differentials of quotients, powers, etc., can then be easily derived — in the following way. The product of x and y, when each is taken as reduced by half of its infinitesimal difference, becomes xy - xdy/2 - ydx/2 + dxdy/4 ; but if x and y are made to increase by the same amount, it becomes xy + xdy/2 + ydx/2 + dxdy/4. Now when the first product is subtracted from the second, ydx + xdy remains as a surplus and this is said to be the surplus of the increase by a whole dx and dy, for this increase is the difference between the two products ; it is therefore the differential of xy. Clearly, in this procedure, the term which forms the chief difficulty, the product of the two infinitesimal differences, cancels itself out. But in spite of the name of Newton it must be said that such an operation although very elementary, is incorrect ; it is not true that (x + dx/2) (y + dy/2) - (x - dx/2) (y - dy/2) = (x + dx) (y + dy) - xy. It can only have been the need to establish the all-important fluxional calculus which could bring a Newton to deceive himself with such a proof.
Other forms which Newton employed in the derivation of differentials are bound up with concrete meanings of the elements and their powers, meanings relating to motion. About the use of the serial form which also characterises his method, it suggests itself to say that it is always possible to obtain the required degree of accuracy by adding more terms and that the omitted terms are relatively insignificant, in general, that the result is only an approximation ; though here too he would have been satisfied with this ground for omission as he is in his method of solving equations of higher degree by approximation, where the higher powers arising from the substitution in the given equation of any ascertained, still inexact term, are omitted on the crude ground of their relative smallness.’
The error into which Newton fell in solving a problem by omitting essential, higher powers, an error which gave his opponents the occasion of a triumph of their method over his, and the true origin of which has been indicated by Lagrange in his recent investigation of it demonstrates the formalism and uncertainty which still prevailed in the use of this instrument. Lagrange shows that Newton made the mistake because he omitted the term of the series containing that power on which the specific problem turned. Newton had kept to the formal, superficial principle of omitting terms on account of their relative smallness. For example, it is well known that in mechanics the terms of the series in which the function of a motion is developed are given a specific meaning, so that the first term or the first function refers to the moment of velocity, the second to the accelerating force and the third to the resistance of forces. Here, then, the terms of the series are not to be regarded merely as parts of a sum, but rather as qualitative moments Of a whole determined by the concept. In this way, the omission of the rest of the terms belonging to the spuriously infinite series acquires an altogether different meaning from omission on the ground of their relative smallness.
[Both considerations are found set simply side by side in the application by Lagrange of the theory of functions to mechanics in the chapter on rectilinear motion The space passed through, considered as a function of the time elapsed, gives the equation x = ft ; this, developed as f(t + d) gives ft + df’t + d2/2.f"t + , etc.
Thus the space traversed in the period of time is represented in the formula as = df’t + d2f"t + d3/2.3f"’t +, etc. The motion by means of which this space has been traversed is (it is said) therefore — i.e. because the analytical development gives several, in fact infinitely, many terms — composed of various partial motions, of which the spaces corresponding to the time will be df’t, d2/2f"t, d3/2.3f"’dt, etc. The first partial — notion is, in known motion, the formally uniform one with a velocity designated by f’t, the second is uniformly accelerated motion derived from an accelerative force proportional to f"t. Now since the remaining terms do not refer to any simple known motion, it is not necessary to take them specially into account and we shall show that they may be abstracted from in determining the motion at the beginning of the point of time.’ This is now shown, but of course only by comparing the series all of whose terms belonged to the determination of the magnitude of the space traversed in the period of time, with the equation given in art. 3 for the motion of a falling body, namely x = at + bt2 in which only these two terms occur. But this equation has itself received this form only because the explanation given to the terms produced by the analytical development is presupposed ; this presupposition is that the uniformly accelerated motion is composed of a formally uniform motion continued with the velocity attained in the preceding period of time, and of an increment (the a in s = at2, i.e. the empirical coefficient) which is ascribed to the force of gravity — a distinction which has no existence or basis whatever in the nature of the thing itself, but is only the falsely physicalised expression of what issues from the assumed analytical treatment.]
The error in the Newtonian solution arose, not because terms of the series were neglected only as parts of a sum, but because the term containing the qualitative determination, which is the essential point, was ignored.
In this example, the procedure is made to depend on the qualitative meaning. In this connection the general assertion can at once be made that the whole difficulty of the principle would be removed if the qualitative meaning of the principle were stated and the operation were made to depend on it — in place of the formalism which links the determination of the differential only to that which gives the problem its name, to the difference as such between a function and its variation after its variable magnitude has received an increment. In this sense, it is obvious that the differential of xn is completely exhausted by the first term of the series which results from the expansion of (x + dx)n . Thus the omission of the rest of the terms is not on account of their relative smallness ; and so there is no assumption of an inexactitude, an error or mistake which could be compensated or rectified by another error — a point of view from which Carnot in particular justifies the ordinary method of the infinitesimal calculus. Since what is involved is not a sum but a relation, the differential is completely given by the first term ; and where further terms, the differentials of higher orders, are required, their determination involves not the continuation of a series as a sum, but the repetition of one and the same relation which alone is desired and which is thus already completely given in the first term. The need for the form of a series, its summation and all that is connected with it, must then be wholly separated from the said interest of the relation.
The explanations of the methods of infinitesimal magnitudes given by Carnot, contain a most lucid exposition of what is essential in the ideas referred to above. But in passing to the practical application itself, there enter more or less the usual ideas about the infinite smallness of the omitted terms relatively to the others. He justifies the method, not by the nature of the procedure itself, but by the fact that the results are correct, and by the advantages of a simplification and shortening of the calculus which follow the introduction of imperfect equations, as he calls them, i.e. those in which such an arithmetically incorrect omission has occurred.
Lagrange, as is well known, reverted to Newton’s original method, that of series, in order to be relieved of the difficulties inherent in the idea of the infinitely small and in the method of first and final ratios and limits. The advantages of his functional calculus as regards precision, abstraction and generality, are sufficiently recognised ; we need mention only what is pertinent here, that it rests on the fundamental principle that the difference, without becoming zero, can be assumed so small that each term of the series is greater than the sum of all the following terms. This method, too, starts from the categories of increment and difference of the function, the variable magnitude of which receives the increment, thereby bringing in the troublesome series of the original function ; also in the sequel the terms to be omitted are considered only as constituting a sum, while the reason for omitting them is made to consist in the relativity of their quantum. And so here, too, on the one hand, the principle of the omission is not brought back to the point of view exemplified in some applications, where (as was remarked above) terms of the series are supposed to have a specific quality significance, and terms are neglected not because of their quantitative insignificance but because they are not qualitatively significant ; and then, on the other hand, the omission itself has no place in the essential point of view which, as regards the so-called differential coefficient, only becomes specifically prominent with Lagrange, in the so-called application of the calculus, as will be more fully considered in the following remark.
The demonstrated qualitative character as such of the form of magnitude here under discussion in what is called the infinitesimal, is found most directly in the category of limit of the ratio referred to above and the carrying out of which in the calculus has been developed into a characteristic method. Lagrange criticises this method as lacking case in application and he claims that the expression limit does not present any definite idea ; this second point we will take up here and examine more closely what is stated about its analytical meaning. Now the idea of limit does indeed imply the true category of the qualitatively determined relation of variable magnitudes above-mentioned ; for the forms of it which occur, dx and dy, are supposed to be taken simply and solely as moments of dy/dx, and dy/dx itself must be regarded as a single indivisible symbol.
That the mechanism of the calculus, especially in its application, thus loses the advantage it derived from the separation of the sides of the differential coefficient, this we will pass over here. Now the said limit is to be limit of a given function ; it is to assign to this function a certain value determined by its mode of derivation. But with the mere category of limit we should not have advanced beyond the scope of this Remark, which is to demonstrate that the infinitely small which presents itself in the differential calculus as dx and dy, does not have merely the negative, empty meaning of a non-finite, non-given magnitude, as when one speaks of ’an infinite multitude’, ’to infinity’, and the like, but on the contrary has the specific meaning of the qualitative nature of what is quantitative, of a moment of a ratio as such. This category, however, merely as such, still has no relation to that which is a given function and does not itself enter into the treatment of such a function or into the use to be made of that determination ; thus the idea of limit, too, confined to this its demonstrated character, would lead nowhere. But the very expression ’limit’ implies that it is a limit of something, i.e. that it expresses a certain value which lies in the function of a variable magnitude ; and we must examine the nature of this concrete role. It is supposed to be the limit of the ratio between the two increments by which the two variable magnitudes connected in an equation (one of which is regarded as a function of the other), are supposed to have been increased ; the increase is taken here as quite undetermined and so far no use is made of the infinitely small. But the way in which this limit is found involves the same inconsistencies as are contained in the other methods. This way is as follows : if y = fx, then when y becomes y + k, fx is to change into fx + ph + qh2 + rh3 and so on ; thus k = ph + qh2, etc., and k/h = p + qh + rh2, etc. Now if k and h vanish, the right-hand side of the equation also vanishes with the exception of p ; now p is supposed to be the limit of the ratio of the two increments. It is clear that while h, as a quantum, is equated with 0, k/h nevertheless is not at the same time equal to 0/0 but is supposed still to remain a ratio.
Now the idea of limit is supposed to have the advantage of avoiding the inconsistency here involved ; p is, at the same time, supposed to be not the actual ratio, which would be 0/0 but only that specific value to which the ratio can infinitely approximate, i.e. can approach so near that the difference can be smaller than any given difference. The more precise meaning of approximation with respect to the terms which are supposed really to approach each other will be considered later. But that a quantitative difference, the definition of which is that it not only can, but shall be smaller than any given difference, is no longer a quantitative difference, this is self-evident, as self-evident as anything can be in mathematics ; but we still have not got away from dy/dx = 0. If on the other hand dy/dx = p, i. e. is assumed to be a definite quantitative ratio as in fact it is, then conversely there is a difficulty about the presupposition which equates h with o, a presupposition which is indispensable for obtaining the equation k/h = p. But if it be granted that k/h = 0, (and when h = 0, k is in fact automatically = 0, for k, the increment of y, depends entirely on the existence of the increment h), then the question would arise, what p — which is a quite definite quantitative value — is supposed to be. To this there is at once an obvious answer, the simple, meagre answer that it is a coefficient derived in such and such a way — the first function, derived in a certain specific manner, of an original function. if we content ourselves with this — and Lagrange did, in fact, do so in practice — then the general part of the science of the differential calculus, and straightway this one particular form of it called the theory of limits would be rid of the increments and of their infinite or arbitrary smallness — spared too, the difficulty of getting rid again of all the terms of a series other than the first, or rather only the coefficient of the first, which inevitably follow on the introduction of these increments ; in addition it would also be purged of those formal categories connected with them, especially of the infinite, of infinite approximation and, too, the categories, here equally empty, of continuous magnitude’ which, moreover, like nisus, becoming, occasion of a variation, are deemed necessary.
[The category of continuous or fluent magnitude enters with the consideration of the external and empirical variation of magnitudes — which are brought by an equation into the relation in which one is a function of the other ; but since the scientific object of the differential calculus is a certain relation (usually expressed by the differential coefficient), the specific nature of which may equally well be called a law, the mere continuity is a heterogeneous aspect of this specific nature, and besides is in any case an abstract and here empty category seeing that nothing whatever is said about the law of continuity. Into what formal definitions one may be led in these matters can be seen from the penetrating exposition by my respected colleague, Prof. Dirksen of the fundamental determinations used in the deduction of the differential calculus, which forms an appendix to the criticism of some recent works on this science. The following definition is actually quoted : ’A continuous magnitude, a continuum, is any magnitude thought of as in a state of becoming such that this becoming takes place not by leaps but by an uninterrupted progress’. This is surely tautologically the same as what was to be defined.]
But it would then be required to show what other meaning and value p has — apart from the meagre definition, quite adequate for the theory, that it is simply a function derived from the expansion of a binomial — i.e. what relationships it embodies and what further use can be made of them mathematically ; this will be the subject of Remark 2. But first we shall proceed to discuss the confusion which the conception of approximation currently used in expositions of the calculus, has occasioned in the understanding of the true, qualitative determinateness of the relation which was the primary interest concerned.
It has been shown that the so-called infinitesimals express the vanishing of the sides of the ratio as quanta, and that what remains is their quantitative relation solely as qualitatively determined ; far from this resulting in the loss of the qualitative relation, the fact is that it is just this relation which results from the conversion of finite into infinite magnitudes. As we have seen, it is in this that the entire nature of the matter consists. Thus in the final ratio, for example, the quanta of abscissa and ordinate vanish ; but the sides of this ratio essentially remain, the one an element of the ordinate, the other an element of the abscissa. This vanishing being represented as ’ an infinite approximation, the previously distinguished ordinate is made to pass over into the other ordinate, and the previously distinguished abscissa into the other abscissa ; but essentially this is not so, the ordinate does not pass over into the abscissa, neither does the abscissa pass into the ordinate. To continue with this example of variable magnitudes, the element of the ordinate is not to be taken as the difference of one ordinate from another ordinate, but rather as the difference or qualitative determination of magnitude relatively to the element 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.
This is the nature of the matter but it is obscured by the fact that what has just been called an element, for example, of the ordinate, is grasped as a difference or increment in such a way that it is only the difference between the quantum of one ordinate and the quantum of another ordinate. And so the limit here does not have the meaning of ratio ; it counts only as the final value to which another magnitude of a similar kind continually approximates in such a manner that it can differ from it by as little as we please, and that the final ratio is a ratio of equality. The infinite difference is thus the fluctuation of a difference of one quantum from another quantum, and the qualitative nature according to which dx is essentially not a determination of the ratio relatively to x, but to dy, comes to be overlooked. Dx is permitted to vanish relatively to dx, but even more does dx vanish relatively to x ; but this means in truth : it has a relation only to dy. In such expositions, geometricians are mainly concerned to make intelligible the approximation of a magnitude to its limit and to keep to this aspect of the difference of quantum from quantum, how it is no difference and yet still is a difference. But all the same, approximation is a category which of itself says nothing and explains nothing ; dx already has approximation behind it ; it is neither near nor nearer ; and ’infinitely near’, itself means the negation of nearness and approximation.
Now since this implies that the increments or infinitesimals have been considered only from the side of the quantum which vanishes in them, and only as a limit, it follows that they are grasped as unrelated moments. From this would follow the inadmissible idea that it is allowed in the final ratio to equate, say abscissa and ordinate, or even sine, cosine, tangent, versed sine, and what not. This idea seems at first to prevail when the arc is treated as a tangent ; for the arc, too, is certainly incommensurable with the straight line, and its element is, in the first place, of another quality than the element of the straight line. It seems even more absurd and inadmissible than the confusing of abscissa, ordinate, versed sine, cosine, etc., when quadrata rotundas, when part of an arc, even though an infinitely small part, is taken to be a part of the tangent and so treated as a straight line. However, this treatment differs essentially from the confusion we have decried ; it is justified by the circumstance that in the triangle which has for its sides the element of an arc and the elements of its abscissa and ordinate, the relation is the same as if this element of the arc were the element of a straight line, of the tangent ; the angles which constitute the essential relation, i.e. that which remains to these elements when abstraction is made from the finite magnitudes belonging to them, are the same. This can also be expressed as the transition of straight lines which are infinitely small, into curved lines, and their relation in their infinity as a relation of curves. Since, according to its definition, a straight line is the shortest distance between two points, its difference from the curved line is based on the determination of amount, on the smaller amount of what is differentiated in this manner, a determination, therefore, of a quantum. But this determination vanishes in the line when it is taken as an intensive magnitude, as an infinite moment, as an element, and with it, too, its difference from the curved line which rested merely on the difference of quantum. As infinite, therefore, the straight line and arc no longer retain any quantitative relation nor consequently, on the basis of the assumed definition, any qualitative difference from each other either ; on the contrary, the former passes into the latter.
Analogous, although also distinct from, the equating of heterogeneous forms is the assumption that infinitely small parts of the same whole are equal to each other ; an assumption in itself indefinite and completely indifferent, but which, applied to an object heterogeneous within itself, i.e. an object whose quantitative determination is essentially non-uniform, produces the peculiar inversion contained in that proposition of higher mechanics which states that infinitely small parts of a curve are traversed in equal, infinitely small times in a uniform motion, inasmuch as this is asserted of a motion in which in equal finite, i.e. existent, parts of time, finite, i.e. existent, unequal parts of the curve are traversed, of a motion therefore which exists as non-uniform and is assumed as such. This proposition is the expression in words of what is supposed to be the significance of an analytical term obtained in the above-mentioned development of the formula relating to a motion which is non-uniform but otherwise conforms to a law. Earlier mathematicians sought to express in words and propositions and to exhibit in geometrical tables the results of the newly invented infinitesimal calculus (which moreover always had to do with concrete objects), chiefly in order to use them for theorems susceptible of the ordinary method of proof. The terms of a mathematical formula into which analytical treatment resolved the magnitude of the object, of motion, for instance, acquired there an objective significance, such as velocity, force of acceleration, and so on ; in accordance with this meaning they were supposed to furnish correct propositions, physical laws ; their objective connections and relations, too, were supposed to be determined in accordance with the analytical connection. A particular example is that in a uniformly accelerated motion there is supposed to exist a special velocity proportional to the times, but that to this velocity there constantly accrues an increment from the force of gravity.
In the modern, analytical form of mechanics such propositions are put forward simply as results of the calculus, without enquiry whether by themselves and in themselves they have a real significance, i.e. one to which there is a corresponding physical existence and whether such meaning can be demonstrated. The difficulty of making intelligible the connection of such forms when they are taken in the real meaning alluded to, for example the transition from said simply uniform velocity to a uniformly accelerated velocity, is held to be completely eliminated by the analytical treatment in which such connection is a simple result of the authority now established once and for all of the operations of the calculus. It is announced as a triumph of science that by means of the calculus alone, laws are found transcending experience, that is, propositions about existence which have no existence. But in the earlier, still naive period of the infinitesimal calculus, the aim was to assign to those forms and propositions represented in geometrical diagrams a real meaning of their own and to make that meaning plausible, and to apply the forms and propositions bearing such meaning in the proof of the main propositions concerned.
It cannot be denied that in this field much has been accepted as proof, especially with the aid of the nebulous conception of the infinitely small, for no other reason than that the result was always already known beforehand, and that the proof which was so arranged that the result did emerge, at least produced the illusion of a framework of proof, an illusion which was still preferred to mere belief or knowledge from experience. But 1 do not hesitate to regard this affectation as nothing more than mere jugglery and window-dressing, and I include in this description even Newton’s proofs, especially those belonging to what has just been quoted, for which Newton has been extolled to the skies and exalted above Kepler, namely that he demonstrated mathematically what Kepler had discovered merely empirically.
The empty scaffolding of such proofs was erected in order to prove physical laws. But mathematics is altogether incapable of proving quantitative determinations of the physical world in so far as they are laws based on the qualitative nature of the moments [of the subject matter] ; and for this reason, that this science is not philosophy, does not start from the Notion, and therefore the qualitative element, in so far as it is not taken lemmatically from experience, lies outside its sphere. The desire to uphold the honour of mathematics, that all its propositions ought to be rigorously proved, has often caused it to forget its limits ; thus it seemed against its honour to acknowledge simply experience as the source and sole proof of empirical propositions. Consciousness has since then developed a more instructed view of the matter ; so long, however, as consciousness is not clearly aware of the distinction between what is mathematically demonstrable and what can come only from another source, between what are only terms of an analytical expansion and what are physical existences, scientific method cannot be developed into a rigorous and pure attitude in this field. Without doubt, however, the same justice will be done to that framework of Newtonian proof as was done to another baseless and artificial Newtonian structure of optical experiments and conclusions derived from them. Applied mathematics is still full of a similar concoction of experiment and reflection ; but just as one part after another of Newtonian optics long since began to be ignored in practice by the science — with the inconsistency however that all the rest although in contradiction was allowed to stand — so, too, it is a fact that already some of those illusory proofs have fallen into oblivion or have been replaced by others.
The Purpose of the Differential Calculus Deduced from its Application
In the previous Remark we considered on the one hand the specific nature of the notion of the infinitesimal which is used in the differential calculus, and on the other the basis of its introduction into the calculus ; both are abstract determinations and therefore in themselves also easy. The so-called application, however, presents greater difficulties, but also the more interesting side ; the elements of this concrete side are to be the object of this Remark. The whole method of the differential calculus is complete in the proposition that dxn = nx(n - 1)dx, or (f(x + i) - fx)/i = P, that is, is equal to the coefficient of the first term of the binomial x + d, or x + 1, developed according to the powers of dx or i. There is no need to learn anything further : the development of the next forms, of the differential of a product, of an exponential magnitude and so on, follows mechanically ; in little time, in half an hour perhaps — for with the finding of the differential the converse the finding of the original function from the differential, or integration, is also given — one can be in possession of the whole theory. What takes longer is simply the effort to understand, to make intelligible, how it is that, after having so easily accomplished the first stage of the task, the finding of the said differential, analytically, i.e. purely arithmetically, by the expansion of the function of the variable after this has received the form of a binomial by the addition of an increment ; how it is that the second stage can be correct, namely the omission of all the terms except the first, of the series arising from the expansion. If all that were required were only this coefficient, then with its determination all that concerns the theory would, as we have said, be settled and done with in less than half an hour and the omission of the further terms of the series (with the determination of the first function, the determination of the second, third, etc., is also accomplished) far from causing any difficulty, would not come into question since they are completely irrelevant.
We may begin by remarking that the method of the differential calculus shows on the face of it that it was not invented and constructed for its own sake. Not only was it not invented for its own sake as another mode of analytical procedure ; on the contrary, the arbitrary omission of terms arising from the expansion of a function is absolutely contrary to all mathematical principles, it being arbitrary in the sense that the whole of this development is nevertheless assumed to belong completely to the matter in hand, this being regarded as the difference between the developed function of a variable (after this has been given the form of a binomial) and the original function. The need for such a mode of procedure and the lack of any internal justification at once suggest that the origin and foundation must lie elsewhere. It happens in other sciences too, that what is placed at the beginning of a science as its elements and from which the principles of the science are supposed to be derived is not self-evident, and that it is rather in the sequel that the raison d’étre and proof of those elements is to be found. The course of events in the history of the differential calculus makes it plain that the matter had its origin mainly in the various so-called tangential methods, in what could be considered ingenious devices ; it was only later that mathematicians reflected on the nature of the method after it had been extended to other objects, and reduced it to abstract formulae which they then also attempted to raise to the status of principle.
We have shown that the specific nature of the notion of the so-called infinitesimal is the qualitative nature of determinations of quantity which are related to each other primarily as quanta ; to this was linked the empirical investigation aimed at demonstrating the presence of this specific nature in the existing descriptions and definitions of the infinitesimal in so far as this is taken as an infinitesimal difference and the like. This was done only in the interest of the abstract nature of the notion as such ; the next question would be as to the nature of the transition from this to the mathematical formulation and application. To this end we must first pursue our examination of the theoretical side, the specific nature of the notion, which will not prove wholly unfruitful in itself ; we must then consider the relation of the theoretical side to its application ; and in both cases we must demonstrate, so far as it is relevant here, that the general conclusions are at the same time adequate to the purpose of the differential calculus and to the way in which the calculus brings about its results.
First, it is to be remembered that the mathematical form of the determinateness of the notion under discussion has already been stated in passing. The specifically qualitative character of quantity is first indicated in the quantitative relation as such ; but it was already asserted in anticipation when demonstrating the so-called kinds of reckoning (see the relative Remark), that it is the relation of powers (still to be dealt with in its proper place) in which number, through the equating of the moments of its Notion, unit and amount, is posited as returned into itself, thereby receiving into itself the moment of infinity, of being-for-self, i.e. of being self-determined. Thus, as we have already said, the express qualitative nature of quantity is essentially connected with the forms of powers, and since the specific interest of the differential calculus is to operate with qualitative forms of magnitude, its own peculiar subject matter must be the treatment of forms of powers, and the whole range of problems, and their solutions, show that the interest lies solely in the treatment of determinations of powers as such.
This foundation is important and at once puts in the forefront something definite in place of the merely formal categories of variable, continuous or infinite magnitudes or even of functions generally ; yet it is still too general, for other operations also have to do with determinations of powers. The raising to a power, extraction of a root, treatment of exponential magnitudes and logarithms, series, and equations of higher orders, the interest and concern of all these is solely with relations which are based on powers. Undoubtedly, these together constitute a system of the treatment of powers ; but which of the various relations in which determinations of powers can be put is the peculiar interest and subject matter of the differential calculus, this is to be ascertained from the calculus itself, i.e. from its so-called applications. These are, in fact, the core of the whole business, the actual procedure in the mathematical solution of a certain group of problems ; this procedure was earlier than the theory or general part and was later called application only with reference to the subsequently created theory, the aim of which was to draw up the general method of the procedure and, as well, to endow it with first principles, i.e. with a justification. We have shown in the preceding Remark the futility of the search for principles which would clarify the method as currently understood, principles which would really solve the contradiction revealed by the method instead of excusing it or covering it up merely by the insignificance of what is here to be omitted (but which really is required by mathematical procedure), or, by what amounts to the same thing, the possibility of infinite or arbitrary approximation and the like. If from the practical part of mathematics known as the differential calculus the general features of the method were to be abstracted in a manner different from that hitherto followed, then the said principles and the concern about them would also show themselves to be superfluous, just as they reveal themselves to be intrinsically false and permanently contradictory.
If we investigate this peculiarity by simply taking up what we find in this part of mathematics, we find as its subject matter :
(a) Equations in which any number of magnitudes (here we can simply confine ourselves to two) are combined into a qualitative whole in such a way that first, these equations have their determinateness in empirical magnitudes which are their fixed limits, and also in the kind of connection they have with these limits and with each other as is generally the case in an equation ; but since there is only one equation for both magnitudes (similarly, relatively more equations for more magnitudes, but always fewer than the number of magnitudes), these equations belong to the class of indeterminate equations ; and secondly, that one aspect of the determinateness of these magnitudes is that they are — or at least one of them is present in the equation in a higher power than the first.
Before proceeding further, there are one or two things to be noticed about this. The first is that the magnitudes, as described under the first of the above two headings, have simply and solely the character of variables such as occur in the problems of indeterminate analysis. Their value is undetermined, but if one of them does receive a completely determined value, i.e. a numerical value, from outside, then the other too, is determined, so that one is a function of the other. Therefore, in relation to the specific quantitative determinateness here in question, the categories of variable magnitudes, functions and the like are, as we have already said, merely formal, because they are still too general to contain that specific element on which the entire interest of the differential calculus is focused, or to permit of that element being explicated by analysis ; they are in themselves simple, unimportant, easy determinations which are only made difficult by importing into them what they do not contain in order that this may then be derived from them — namely, the specific determination of the differential calculus. Then as regards the so-called constant, we can note that it is in the first place an indifferent empirical magnitude determining the variables only with respect to their empirical quantum as a limit of their minimum and maximum ; but the nature of the connection between the constants and the variables is itself a significant factor in the nature of the particular function which these magnitudes are. Conversely, however, the constants themselves are also functions ; in so far as a straight line, for example, has the meaning of being the parameter of a parabola, then this meaning is that it is the function y2/x2 ; and in the expansion of the binomial generally, the constant which is the coefficient of the first term of the development is the sum of the roots, the coefficient of the second is the sum of the products, in pairs, and so on ; here, therefore, the constants are simply functions of the roots. Where, in the integral calculus, the constant is determined from the given formula, it is to that extent treated as a function of this. Further on we shall consider these coefficients in another character than that of functions, their meaning in the concrete object being the focus of the whole interest.
Now the difference between variables as considered in the differential calculus, and in their character as factors in indeterminate problems, must be seen to consist in what has been said, namely, that at least one of those variables (or even all of them), is found in a power higher than the first ; and here again it is a matter of indifference whether they are all of the same higher power or are of unequal powers ; their specific indeterminateness which they have here consists solely in this, that in such a relation of powers they are functions of one another. The alteration of variables is in this way qualitatively determined, and hence continuous, and this continuity, which again is itself only the purely formal category of an identity, of a determinateness which is preserved and remains self-same in the alteration, has here its determinate meaning, solely, that is, in the power-relation, which does not have a quantum for its exponent and which forms the non-quantitative, permanent determinateness of the ratio of the variables. For this reason it should be noted, in criticism of another formalism, that the first power is only a power in relation to higher powers ; on its own, x is merely any indeterminate quantum. Thus there is no point in differentiating for their own sakes the equations y = ax + b (of the straight line), or s = ct (of the plain uniform velocity) ; if from y = ax, or even ax + b, we obtain a = dy/dx, or from s = ct, ds/dt = c, then a = y/x is equally the determination of the tangent, or s/t that of velocity simply as such. The latter is given the form of dy/dx in the context of what is said to be the development of the uniformly accelerated motion ; but, as already remarked, the presence in the system of such a motion, of a moment of simple, merely uniform velocity, i.e. a velocity which is not determined by the higher power of one of the moments of the motion is itself an empty assumption based solely on the routine of the method. Since the method starts from the conception of the increment which the variable is supposed to acquire, then of course a variable which is only a function of the first power can also receive an increment ; when now in order to find the differential we have to subtract the difference of the second equation thus produced from the given equation, the meaninglessness of the operation becomes apparent, for, as we have remarked, the equation for the so-called increments, both before and after the operation, is the same as for the variables themselves.
(b) What has been said determines the nature of the equation which is to be treated ; we have now to indicate what is the interest on which the treatment of the equation is focused. This consideration can yield only known results, in a form found especially in Lagrange’s version ; but I have made the exposition completely elementary in order to eliminate the heterogeneous determinations associated with it. The basis of treatment of an equation of this kind shows itself to be this, that the power is taken as being within itself a relation or a system of relations. We said above that power is number which has reached the stage where it determines its own alteration, where its moments of unit and amount are identical — as previously shown, completely identical first in the square, formally (which makes no difference here) in higher powers. Now power is number (magnitude as the more general term may be preferred, but it is in itself always number), and hence a plurality, and also is represented as a sum ; it can therefore be directly analysed into an arbitrary amount of numbers which have no further determination relatively to one another or to their sum, other than that together they are equal to the sum. But the power can also be split into a sum of differences which are determined by the form of the power. If the power is taken as a sum, then its radical number, the root, is also taken as a sum, and arbitrarily after manifold divisions, which manifoldness, however, is the indifferent, empirically quantitative element. The sum which the root is supposed to be, when reduced to its simple determinateness, i.e. to its genuine universality, is the binomial ; all further increase in the number of terms is a mere repetition of the same determination and therefore meaningless.
[It springs solely from the formalism of that generality to which analysis perforce lays claim when, instead of taking (a + b)n for the expansion of powers, it gives the expression the form of (a + b + c + d...)n as happens too in many other cases ; such a form is to be regarded as, so to speak, a mere affectation of a show of generality ; the matter itself is exhausted in the binomial. It is through the expansion of the binomial that the law is found, and it is the law which is the genuine universality, not the external, mere repetition of the law which is all that is effected by this a + b + c + d ...]
The sole point of importance here is the qualitative determinateness of the terms resulting from the raising to a power of the root taken as a sum, and this determinateness lies solely in the alteration which the potentiation is. These terms, then, are wholly functions of potentiation and of the power. Now this representation of number as a sum of a plurality of terms which are functions of potentiation, and the finding of the form of such functions and also this sum from the plurality of those terms, in so far as this must depend solely on that form, this constitutes, as we know, the special theory of series. But in this connection it is essential to distinguish another object of interest, namely the relation of the fundamental magnitude itself (whose determinateness, since it is a complex, i.e. here an equation, includes within itself a power) to the functions of its potentiation. This relation, taken in complete abstraction from the previously mentioned interest of the sum, will show itself to be the sole standpoint yielded by the practical aspect of the science.
But first, another determination must be added to what has been said, or rather, one which is implied in it must be removed. It was said that the variable into the determination of which power enters is regarded as within itself a sum, in fact a system of terms in so far as these are functions of the potentiation, and that thus the root, too, is regarded as a sum and in the simply determined form of a binomial : xn = (y + z)n = (yn + ny(n-1)z + ... ). This exposition started from the sum as such for the expansion of the power, i.e. for obtaining the functions of its potentiation ; but what is concerned here is not a sum as such, or the series arising from it ; what is to be taken up from the sum is only the relation. The relation as such of the magnitudes is, on the one hand, all that remains after abstraction is made from the plus of a sum as such, and on the other hand, all that is needed for finding the functions produced by the expansion of the power.
But such relation is already determined by the fact that here the object is an equation, ym = axn, and so already a complex of several (variable) magnitudes which contains a power determination of them. In this complex, each of these variables is posited simply as in relation to the others with the meaning, one could say, of a plus implicit in it — as a function of the other variables ; their character, that of being functions of one another, gives them this determination of a plus which, however, for that same reason, is wholly indeterminate — not an increase or an increment, or anything of that nature. Yet even this abstract point of view we could leave out of account ; we can quite simply stop at the point where the variables in the equation having received the form of functions of one another, such functions containing a relation of powers, the functions of potentiation are then also compared with one another — these second functions being determined simply and solely by the potentiation itself. To treat an equation of the powers of its variables as a relation of the functions developed by potentiation can, in the first place, be said to be just a matter of choice or a possibility ; the utility of such a transformation has to be indicated by some further purpose or use ; and the sole reason for the transformation was its utility.
When we started above from the representation of these functions of potentiation of a variable which is taken as a sum complex within itself, this served only partly to indicate the nature of such functions, but partly also to show the way in which they are found.
What we have here then is the ordinary analytical development which for the purpose of the differential calculus is operated in this way, that an increment dx or i is given to the variable and then the power of the binomial is developed by the terms of the series belonging to it. But the so-called increment is supposed to be not a quantum but only a form, the whole value of which is that it assists the development ; it is admitted — most categorically by Euler and Lagrange and in the previously mentioned conception of limit — that what is wanted is only the resulting power determinations of the variables, the so-called coefficients, namely, of the increment and its powers, according to which the series is ordered and to which the different coefficients belong. On this we could perhaps remark that since an increment (which has no quantum) is assumed only for the sake of the development, it would be most appropriate to take i (the one) for that purpose, for in the development this always occurs only as a factor ; the factor one, therefore, fulfils the purpose, namely, that the increment is not to involve any quantitative determinateness or alteration ; on the other hand, dx, which is burdened with the false idea of a quantitative difference, and other symbols like i with the mere show — pointless here — of generality, always have the appearance and pretension of a quantum and its powers ; which pretension then involves the trouble that they must nevertheless be removed and left out. In order to retain the form of a series expanded on the basis of powers, the designations of the exponents as indices could equally well be attached to the one. But in any case, abstraction must be made from the series and from the determination of the coefficients according to their place in the series ; the relation between all of them is the same ; the second function is derived from the first in exactly the same manner as this is from the original function, and for the function counted as second, the first derived function is itself original. But the essential point of interest is not the series but simply and solely the determination of the power resulting from the expansion in its relation to the variable which for the power determination is immediate. It should not therefore be defined as the coefficient of the first term of the development, for it is first only in relation to the other terms following it in the series, and a power such as that of an increment, like the series itself, is here out of place ; instead, the simple expression : derived function of a power, or as was said above : function of potentiation of a magnitude, would be preferable — the knowledge of the way in which the derivation is taken to be a development included within a power being presupposed.
Now if the strictly mathematical beginning in this part of analysis is nothing more than the finding of the function determined by the expansion of the power, the further question is what is to be done with the relation so obtained, where has it an application and use, or indeed, for what purpose are such functions sought. It is the finding of relations in a concrete subject matter which can be reduced to such a function that has given the differential calculus its great interest.
But as regards the applicableness of the relation, we need not wait for conclusions to be drawn from particular applications themselves, the answer follows directly and automatically from the nature of the matter which we have shown to consist in the form possessed by the moments of powers : namely, the expansion of the powers, which yields the functions of their potentiation, contains (ignoring any more precise determination) in the first place, simply the reduction of the magnitude to the next lower power. This operation is therefore applicable in the case of those objects in which there is also present such a difference of power determinations. Now if we reflect on the specific nature of space, we find that it contains the three dimensions which, in order to distinguish them from the abstract differences of height, length and breadth, we can call concrete — namely, line, surface and total space ; and when they are taken in their simplest forms and with reference to self-determination and consequently to analytical dimensions, we have the straight line, plane surface and surface taken as a square, and the cube. The straight line has an empirical quantum, but with the plane there enters the qualitative element, the determination of power ; further modifications, e.g. the fact that this also happens in the case of plane curves, we need not consider, for we are concerned primarily with the distinction in general. With this there arises, too, the need to pass from a higher power to a lower, and vice versa, when, for example, linear determinations are to be derived from given equations of the plane, or vice versa. Further, the motion in which we have to consider the quantitative relation of the space traversed to the time elapsed, manifests itself in the different determinations of a motion which is simply uniform, or uniformly accelerated, or alternately uniformly accelerated and uniformly retarded, and thus a self-returning motion ; since these different kinds of motion are expressed in accordance with the quantitative relation of their moments, of space and time, their equations contain different determinations of powers, and when it is necessary to determine one kind of motion, or a spatial magnitude to which one kind of motion is linked, from another kind of motion, the operation also involves the passage from one power-function to another, either higher or lower. These two examples may suffice for the purpose for which they are cited.
The appearance of arbitrariness presented by the differential calculus in its applications would be clarified simply by an awareness of the nature of the spheres in which its application is permissible and of the peculiar need for and condition of this application. But now the further point of interest within these spheres themselves is to know between what parts of the subject matter of the mathematical problem such a relation occurs as is posited peculiarly by the differential calculus. First, it must be observed that there are two kinds of relation. The operation of depotentiating an equation considered according to the derivative functions of its variables, yields a result which, in itself, is no longer truly an equation but a relation ; this relation is the subject matter of the differential calculus proper. This also gives us, secondly, the relation of the higher power form (the original equation) itself to the lower (the derivative). This second relation we must ignore for the time being ; it will prove to be the special subject matter of the integral calculus.
Let us start by considering the first relation ; for the determination of its moment (to be taken from the application, in which lies the interest of the operation) we shall take the simplest example from curves determined by an equation of the second degree. As we know, the relation of the co-ordinates is given directly by the equation in a power form. From the fundamental determination follow the determinations of the other straight lines connected with the co-ordinates, tangent, subtangent, normal, and so on.
But the equations between these lines and the co-ordinate are linear equations ; the wholes with respect to which these lines are determined as parts, are right-angled triangles formed by straight lines. The transition from the original equation which contains the power form, to said linear equations, involves now the above-mentioned transition from the original function (which is an equation), to the derived function (which is a relation, a relation, that is, between certain lines contained in the curve). The problem consists in finding the connection between the relation of these lines and the equation of the curve.
It is not without interest, as regards the historical element, to remark this much, that the first discoverers could only record their findings in a wholly empirical manner without being able to account for the operation, which remained a completely external affair. It will be sufficient here to refer to Barrow, to him who was Newton’s teacher. In his lect. Opt. et Geom., in which he treats problems of higher geometry according to the method of indivisibles, a method which, to begin with, is distinct from the characteristic feature of the differential calculus, he also puts on record’ his procedure for determining tangents — ’because his friends urged him to do so’. To form a proper idea of how this procedure is formulated simply as an external rule, in the same style as the ’rule of three’, or better still the so-called ’test by casting out nines’, one must read Barrow’s own exposition. He draws the tiny lines afterwards known as the increments in the characteristic triangle of a curve and then gives the instruction, in the form of a mere rule, to reject as superfluous the terms which, as a result of the expansion of the equations, appear as powers of the said increments or as products (etenim isti termini nihilum valebunt) ; similarly, the terms which contain only magnitudes to be found in the original equation are to be rejected (the subsequent subtraction of the original equation from that formed with the increments) ; and finally, for the increments of the ordinate and abscissa, the ordinate itself and the subtangent respectively are to be substituted. The procedure, if one may say so, can hardly be set forth in a more schoolmaster-like manner ; the latter substitution is the assumption of the proportionality of the increments of the ordinate and the abscissa with the ordinate and the subtangent, an assumption on which is based the determination of the tangent in the ordinary differential method ; in Barrow’s rule this assumption appears in all its naive nakedness. A simple way of determining the subtangent was found ; the artifices of Roberval and Fermat have a similar character. The method for finding maximal and minimal values from which Fermat started rests on the same basis and the same procedure. It was a mathematical craze of those times to find so-called methods, i.e. rules of that kind and to make a secret of them — which was not only easy, but in one respect even necessary, for the same reason that it was easy — namely, because the inventors had found only an empirical, external rule, not a method, i.e. nothing derived from established principles. Leibniz accepted such so-called methods from his contemporaries and so did Newton who got them directly from his teacher ; by generalising their form and applicableness they opened up new paths for the sciences, but at the same time they also felt the need to wrest free the procedure from the shape of merely external rules and to try to procure for it the necessary justification.
If we analyse the method more closely, we find the genuine procedure to be as follows. Firstly the power forms (of the variables of course) contained in the equation are reduced to their first functions. But the value of the terms of the equation is thereby altered ; there is now no longer an equation, but instead only a relation between the first function of the one variable and the first function of the other. Instead of px = y2 we have p : 2y, or instead of 2ax - x2 = y2, we have a - x : y, the relation which later came to be designated dy/dx. Now the equation represents a curve ; but this relation, which is completely dependent on it and derived from it (above, according to a mere rule), is, on the contrary, a linear relation with which certain lines are in proportion : p : 2y or a - x : y are themselves relations of straight line of the curve, of the co-ordinates and parameters. But with all this, nothing is as yet known. The interest centres on finding that the derived relation applies to other lines connected with the curve, on finding the equality of two relations. And so there is, secondly, the question, which are the straight lines determined by the nature of the curve, standing in such a relation ? But this is just what was already known : namely, that the relation so obtained is the relation of the ordinate to the subtangent. This the ancients had found in an ingenious geometrical manner ; what the moderns have discovered is the empirical procedure of so preparing the equation of the curve that it yields that first relation of which it was already known that it is equal to a relation containing the line (here the subtangent) which is to be determined. Now on the one hand, this preparation of the equation — the differentiation — has been methodically conceived and executed ; but on the other hand the imaginary increments of the co-ordinates and an imaginary characteristic triangle formed by them and by an equally imaginary increment of the tangent, have been invented in order that the proportionality of the ratio found by lowering the degree of the equation to the ratio formed by the ordinate and subtangent, may be represented, not as something only empirically accepted as an already familiar fact, but as something demonstrated. However, in the said form of rules, the already familiar fact reveals itself absolutely and unmistakably as the sole occasion and respective justification of the assumption of the characteristic triangle and the said proportionality.
Now Lagrange rejected this pretence and took the genuinely scientific course. We have to thank his method for bringing into prominence the real point of interest for it consists in separating the two transitions necessary for the solution of the problem and treating and proving each of them separately. One part of this solution (for the more detailed statement of the process we shall confine ourselves to the example of the elementary problem of finding the subtangent), the theoretical or general part, namely, the finding of the first function from the given equation of the is dealt with separately ; the result is a linear relation, a curve, relation therefore of straight lines occurring in the system determined by the curve. The other part of the solution now is the finding of those lines in the curve which stand in this relation. Now this is effected in a direct manner i.e., without the characteristic triangle, which means that there is no assumption of infinitely small arcs, ordinates and abscissae, the last two being given the significance of dy and dx, that is, of being sides of that relation, and at the same time directly equating the infinitely small ordinate and abscissa with the ordinate and subtangent themselves. A line (and a point, too), is determined only in so far as it forms the side of a triangle and the determination of a point, too, falls only in such triangle. This, it may be mentioned in passing, is the fundamental proposition of analytical geometry from which are derived the co-ordinates of that science, just as (it is the same standpoint) in mechanics it gives rise to the parallelogram of forces, for which very reason the many efforts to find a proof of this latter are quite unnecessary. The subtangent, now, is made to be the side of a triangle whose other sides are the ordinate and the tangent connected to it. The equation of the latter, as a straight line, is p = aq (the determination does not require the additional term, + b which is added only on account of the fondness for generality) ; — the determination of the ratio p/q falls within a, the coefficient of q which is the respective first function (derivative) of the equation, but may simply be considered only as a = p/q being, as we have said, the essential determination of the straight line which is applied as tangent to the curve. But the first function (derivative) of the equation of the curve is equally the determination of a straight line ; seeing then that the co-ordinate p of the first straight line and y, the co-ordinate of the curve, are assumed to be identical (so that the point at which the curve is touched by the first straight line assumed as tangent is also the starting point of the straight line determined by the first function of the curve), the problem is to show that this second straight line coincides with the first, i.e. is a tangent ; or, algebraically expressed, that since y = fx and p = Fq, and it is assumed that y = p and hence that fx = Fq, therefore f’x = F’q. Now in order to show that the straight line applied as a tangent and the straight line determined by the first function of the equation coincide, and that therefore the latter is a tangent, Descartes has recourse to the increment i of the abscissa and to the increment of the ordinate determined by the expansion of the function. Thus here, too, the objectionable increment also makes its appearance ; but its introduction for the purpose indicated and its role in the expansion of the function must be carefully distinguished from the previously mentioned employment of the increment in finding the differential equation and in the characteristic triangle. Its employment here is justified and necessary because it falls within the scope of geometry, the geometrical determination of a tangent as such implying that between it and the curve with which it has a point in common, no other straight line can be drawn which also passes through the said point. For, as thus determined, the quality of tangent or not-tangent is reduced to a quantitative difference, that line being the tangent of which simply greater smallness is predicated with respect to the determination in point. This seemingly only relative smallness contains no empirical element whatever, i.e. nothing dependent on a quantum as such ; in virtue of the nature of the formula it is explicitly qualitative if the difference of the moments on which the magnitude to be compared depends is a difference of powers. Since this difference becomes that of i and i2 and i (which after all is meant to signify a number) is then to be conceived as a fraction, i2 is therefore in itself and explicitly smaller than i, so that the very conception of an arbitrary magnitude in connection with i is here superfluous and in fact out of place. For the same reason the demonstration of the greater smallness has nothing to do with an infinitesimal, which thus need not be brought in here at all.
I must also mention the tangential method of Descartes, if only for its beauty and its fame — well-deserved but nowadays mostly forgotten ; it has, moreover, a bearing on the nature of equations and this, again, calls for a further remark. Descartes expounds this independent method, in which the required linear determination is likewise found from the same derivative function, in his geometry which has proved to be so fruitful in other respects tool ; in it he has taught the great basis of the nature of equations and their geometrical construction, and also of the application of analysis, thereby greatly widened in its scope, to geometry. With him the problem took the form of drawing straight lines perpendicularly to given points on a curve as a method for determining the subtangent, etc. One can understand the satisfaction he felt at his discovery, which concerned an object of general scientific interest at that time and which is so purely geometrical and therefore was greatly superior to the mere rules of his rivals, referred to above. His words are as follows : ’J’ose dire que c’est ceci le probleme le plus utile et le plus general, non seulement que je sache, mais meme que j’aie l’amais desire de savoir en giometrie.’ He bases his solution on the analytic equation of the right-angled triangle formed by the ordinate of the point on the curve to which the required straight line in the problem is to be drawn perpendicularly, by this same straight line (the normal), and thirdly, by that part of the axis which is cut off by the ordinate and the normal (the subnormal). Now from the known equation of a curve, the value of either the ordinate or the abscissa is substituted in the said triangle, the result being an equation of the second degree (and Descartes shows how even curves whose equations contain higher powers reduce to this) ; in this equation, only one of the variables occurs, namely, as a square and in the first degree — a quadratic equation which at first appears as a so-called impure equation. Descartes now makes the reflection that if the assumed point on the curve is imagined to be a point of intersection of the curve and of a circle, then this circle will also cut the curve in another point and we shall then get for the unequal xs thus produced, two equations with the same constants and of the same form, or else only one equation with unequal values of x. But the equation only becomes one for the one triangle in which the hypotenuse is perpendicular to the curve or is the normal, the case being conceived of in this way, that the two points of intersection of the curve and the circle are made to coincide and the circle is thus made to touch the curve. But in that case it is also true that the x or y of the quadratic equation no longer have unequal roots. Now since in a quadratic equation with two equal roots the coefficient of the term containing the unknown in the first power is twice the single root, we obtain an equation which yields the required determinations. This procedure must be regarded as the brilliant device of a genuinely analytical mind, in comparison with which the dogmatically assumed proportionality of the subtangent and the ordinate with the postulated infinitely small, so-called increments, of the abscissa and ordinate drops into the background.
The final equation obtained in this way, in which the coefficient of the second term of the quadratic equation is equated with the double root or unknown, is the same as that obtained by the method of the differential calculus. The differentiation of x2 - ax - b = 0 yields the new equation 2x - a = 0 ; or x3 - px - q = 0 gives 3x2 - p = 0. But it suggests itself here to remark that it is by no means self-evident that such a derivative equation is also correct. We have already pointed out that an equation with two variables (which, just because they are variables, do not lose their character of being unknown quantities) yields only a proportion ; and for the simple reason stated, namely, that when the functions of potentiation are substituted for the powers themselves, the value of both terms of the equation is altered and it is not yet known whether an equation still exists between them with their values thus altered. All that the equation dy/dx = P expresses is that P is a ratio and no other real meaning can be ascribed to dyldx. But even so, we still do not know of this ratio = P, to what other ratio it is equal ; and it is only such equation or proportionality which gives a value and meaning to it. We have already mentioned that this meaning, which was called the application, was taken from another source, empirically ; similarly, in the case of the equations here under discussion which have been obtained by differentiation, it is from another source that we must know whether they have equal roots in order that we may learn whether the equation thus obtained is still correct. But this fact is not expressly brought to notice in the textbooks ; it is disposed of, certainly, when an equation with one unknown, reduced to zero, is straightway equated with y, with the result, of course, that differentiation yields a dy/dx, i.e. only a ratio. The functional calculus, it is true, is supposed to deal with functions of potentiation and the differential calculus with differentials ; but it by no means follows from this alone that the magnitudes from which the differentials or functions of potentiation are taken, are themselves supposed to be only functions of other magnitudes. Besides, in the theoretical part, in the instruction to derive the differentials, i.e. the functions of potentiation, there is no indication that the magnitudes which are to be subjected to such treatment are themselves supposed to be functions of other magnitudes.
Further, with regard to the omission of the constant when differentiating, we may draw attention to the fact that the omission has here the meaning that the constant plays no part in the determination of the roots if these are equal, the determination being exhausted by the coefficient of the second term of the equation : as in the example quoted from Descartes where the constant is itself the square of the roots, which therefore can be determined from the constant as well as from the coefficients — seeing that, like the coefficients, the constant is simply a function of the roots of the equation. In the usual exposition, the omission of the so-called constants (which are connected with the other terms only by plus and minus) results from the mere mechanism of the process of differentiation, in which to find the differential of a compound expression only the variables are given an increment, and the expression thereby formed is subtracted from the original expression. The meaning of the constants and of their omission, in what respect they are themselves functions and, as such, are or are not of service, are not discussed.
In connection with the omission of constants we may make a similar observation about the names of differentiation and integration as we did before about the expressions finite and infinite : that is, that the character of the operation in fact belies its name. To differentiate denotes that differences are posited, whereas the result of differentiating is, in fact, to reduce the dimensions of an equation, and to omit the constant is to remove from the equation an element in its determinateness. As we have remarked, the roots of the variables are made equal, and therefore their difference is cancelled. In integration, on the other hand, the constant must be added in again and although as a result the equation is integrated, it is so in the sense that the previously cancelled difference of the roots is restored, that is, what was posited as equal is differentiated again. The ordinary expression helps to obscure the essential nature of the matter and to set everything in a point of view which is not only subordinate but even alien to the main interest, the point of view, namely, of the infinitely small difference, the increment and the like, and also of the mere difference as such between the given and the derived function, without any indication of their specific, i.e. qualitative, difference.
Another important sphere in which the differential calculus is employed is mechanics. The meanings of the distinct power functions yielded by the elementary equations of its subject matter, motion, have already been mentioned in passing ; at this point, I shall proceed to deal with them directly. The equation, i.e. the mathematical expression, for simply uniform motion, c = s/t or s = ct, in which the spaces traversed are proportional to the times elapsed in accordance with an empirical unit c (the magnitude of the velocity), offers no meaning for differentiation : the coefficient c is already completely determined and known, and no further expansion of powers is possible. We have already noticed how s = at2, the equation of the motion of a falling body, is analysed ; the first term of the analysis, ds/dt = 2at is translated into language, and also into existence, in such a manner that it is supposed to be a factor in a sum (a conception we have long since abandoned), to be one part of the motion, which part moreover is attributed to the force of inertia, i.e. of a simply uniform motion, in such a manner that in infinitely small parts of time the motion is uniform, but in finite parts of time, i.e. in actually existent parts of time, it is non-uniform. Admittedly, fs = 2at ; and the meaning of a and t themselves is known and so, too, the fact that the motion is determined as of uniform velocity ; since a = s/t2, 2at is equal simply to 2s/t.
But knowing this we are not a whit wiser ; it is only the erroneous assumption that 2at is a part of the motion regarded as a sum, that gives the false appearance of a physical proposition. The factor itself, a, the empirical unit — a simple quantum — is attributed to gravity ; but if the category of ’force of gravity’ is to be employed then it ought rather to be said that the whole, s = at2, is the effect, or, better, the law, of gravity. Similarly with the proposition derived from ds/dt = 2at, that if gravity ceased to act, the body, with the velocity reached at the end of its fall, would cover twice the distance it had traversed, in the same period of time as its fall. This also implies a metaphysics which is itself unsound : the end of the fall, or the end of a period of time in which the body has fallen, is itself still a period of time ; if it were not, there would be assumed a state of rest and hence no velocity, for velocity can only be fixed in accordance with the space traversed in a period of time, not at its end. When, however, the differential calculus is applied without restriction in other departments of physics where there is no motion at all, as for example in the behaviour of light (apart from what is called its propagation in space) and in the application of quantitative determinations to colours, and the first function of a quadratic function here is also called a velocity, then this must be regarded as an even more illegitimate formalism of inventing an existence.
The motion represented by the equation s = at2 we find, says Lagrange, empirically in falling bodies ; the next simplest motion would be that whose equation were s = ct3, but no such motion is found in Nature ; we do not know what significance the coefficient c could have. Now though this is indeed the case, there is nevertheless a motion whose equation is s3 = at2 — Kepler’s law of the motion of the bodies of the solar system ; the significance 2at here of the first derived function 2at/3s2 and the further direct treatment of this equation by differentiation, the development of the laws and determinations of that absolute motion from this starting point, must indeed present an interesting problem in which analysis would display a brilliance most worthy of itself.
Thus the application of the differential calculus to the elementary equations of motion does not of itself offer any real interest ; the formal interest comes from the general mechanism of the calculus. But another significance is acquired by the analysis of motion in connection with the determination of its trajectory ; if this is a curve and its equation contains higher powers, then transitions are required from rectilinear functions, as functions of potentiation, to the powers themselves ; and since the former have to be obtained from the original equation of motion containing the factor of time, this factor being eliminated, the powers must at the same time be reduced to the lower functions of development from which the said linear equations can be obtained. This aspect leads to the interesting feature of the other part of the differential calculus.
The aim of the foregoing has been to make prominent and to establish the simple, specific nature of the differential calculus and to demonstrate it in some elementary examples. Its nature has been found to consist in this, that from an equation of power functions the coefficient of the term of the expansion, the so-called first function, is obtained, and the relation which this first function represents is demonstrated in moments of the concrete subject matter, these moments being themselves determined by the equation so obtained between the two relations. We shall also briefly consider the principle of the integral calculus to see what light is thrown on its specific, concrete nature by the application of the principle. The view of the integral calculus has been simplified and more correctly determined merely by the fact that it is no longer taken to be a method of summation in which it appeared essentially connected with the form of series ; the method was so named in contrast to differentiation where the increment counts as the essential element. The problem of this calculus is, in the first instance, like that of the differential calculus, theoretical or rather formal, but it is, as everyone knows, the converse of the latter. Here, the starting point is a function which is considered as deriv,ed, as the coefficient of the first term arising from the expansion of an equation as yet unknown, and the problem is to find the original power function from the derivative ; what would be regarded in the natural order of the expansion as the original function is here derived, and the function previously regarded as derived is here the given, or simply original, function. Now the formal part of this operation seems to have been accomplished already in the differential calculus in which the transition and the relation of the original to the derived function in general has been established. Although in doing this it is necessary in many cases to have recourse to the form of series simply in order to obtain the function which is to be the starting point and also to effect the transition from it to the original function, it is important to remember that this form as such has nothing directly to do with the peculiar principle of integration.
The other part of the problem of the calculus appears in connection with its formal operation, namely the application of the latter. But this now is itself the problem : namely, to find the meaning in the above-mentioned sense, possessed by the original function of the given function (regarded as first) of a particular subject matter ; it might seem that this doctrine, too, was in principle already finally settled in the differential calculus ; but a further circumstance is involved which prevents the matter from being so simple. In the differential calculus, namely, it was found that the linear relation is obtained from the first function of the equation of a curve, so that it is also known that the integration of this relation gives the equation of the curve in the relation of abscissa and ordinate ; or, if the equation for the area enclosed by the curve were given, then we should be supposed to know already from the differential calculus that the meaning of the first function of such equation would be that it represented the ordinate as a function of the abscissa, and therefore the equation of the curve.
The problem now is to determine which of the moments determining the subject matter is given in the equation itself ; for the analytical treatment can only start from what is given and then pass on to the other moments of the subject matter. What is given is, for example, not the equation of an area enclosed by the curve, nor, say, of the figure resulting from its rotation ; nor again of an arc of the curve, but only the relation of the abscissa and ordinate in the equation of the curve itself. Consequently, the transitions from those determinations to this equation itself cannot yet be dealt with in the differential calculus ; the finding of these relations is reserved for the integral calculus.
But further, it has been shown that the differentiation of an equation of several variables yields the derived function or differential coefficient, not as an equation but only in the form of a ratio ; the problem is then to find in the moments of the given subject matter a second ratio that is equal to this first ratio which is the derived function. By contrast, the object of the integral calculus is the relation itself of the original to the derived function, which latter is here supposed to be given ; so that the problem concerns the meaning to be assigned to the sought-for original function in the subject matter of the given first derived function ; or rather, since this meaning, for example, the area enclosed by a curve or the rectification of a curve represented as a straight line, already finds expression in the statement of the problem, to show that an original function has that meaning, and which is the moment of the subject matter which must be assumed for this purpose as the initial function of the derived function.
Now the usual method makes the matter easy for itself by using the idea of the infinitesimal difference ; for the quadrature of curves, an infinitely small rectangle, a product of the ordinate into the element, i.e. the infinitesimal bit of the abscissa, is taken for the trapezium one of whose sides is the infinitely small arc opposite to the infinitesimal bit of the abscissa ; the product is now integrated in the sense that the integral is the sum of the infinitely many trapezia or the area to be determined — namely, the finite magnitude of this element of the area. Similarly, from an infinitely small element of the arc and the corresponding ordinate and abscissa, the ordinary method forms a right-angled triangle in which the square of the arc element is supposed to be equal to the sum of the squares of the two other infinitely small elements, the integration of this giving the length of the arc itself as a finite quantity.
This procedure rests on the general discovery on which this field of analysis is based, in this instance, namely, that the quadrated curve, or the rectified arc, stands to a certain function given by the equation of the curve, in the relation of the so-called original function to its derivative. The aim of the integral calculus is this : when a certain part of a mathematical object (e.g. of a curve) is assumed to be the derived function, which other part of the object is expressed by the corresponding original function ? It is known that when the function of the ordinate given by the equation of the curve is taken as the derived function, the corresponding original function gives the quantitative expression for the area of the curve cut off by this ordinate ; and, when a certain tangential determination is identified with the derived function, the corresponding original function expresses the length of the arc belonging to this tangential determination, and so on. But the method which employs the infinitesimal, and operates with it mechanically, simply makes use of the discovery that these relations — the one of an original function to its derivative and the other of the magnitudes of two parts or elements of the mathematical object — form a proportion, and spares itself the trouble of demonstrating the truth of what it simply presupposes as a fact. The singular merit here of mathematical acumen is to have found out from results already known elsewhere, that certain specific aspects of a mathematical object stand in the relationship to each other of the original to the derived function.
Of these two functions it is the derived function or, as it has been defined, the function of potentiation, which here in the integral calculus is given relatively to the original, which has first to be found by integration. But the derived function is not directly given, nor is it at once evident which part or element of the mathematical object is to be correlated with the derived function in order that by reducing this to the original function there may be found that other part or element, whose magnitude is required to be determined. The usual method, as we have said, begins by representing certain parts of the object as infinitely small in the form of derived functions determinable from the originally given equation of the object simply by differentiation (like the infinitely small abscissae and ordinates in connection with the rectification of a curve) ; the parts selected are those which can be brought into a certain relation (one established in elementary mathematics) with the subject matter of the problem (in the given example, with the arc) this, too, being represented as infinitely small, and from this relation the magnitude required to be known can be found from the known magnitude of the parts originally taken. Thus, in connection with the rectification of curves, the three infinitely small elements mentioned are connected in the equation of the right-angled triangle, while for the quadrature of curves, seeing that area is taken arithmetically to be simply the product of lines, the ordinate and the infinitely small abscissa are connected in the form of a product. The transition from such so-called elements of the area, the arc, etc., to the magnitude of the total area or the whole arc itself, passes merely for the ascent from the infinite expression to the finite expression, or to the sum of the infinitely many elements of which the required magnitude is supposed to consist.
It is therefore merely superficial to say that the integral calculus is simply the converse, although in general the more difficult, problem of the differential calculus ; the real interest of the integral calculus concerns almost exclusively the relation between the original and the derived function in the concrete subject matter.
Even in this part of the calculus, Lagrange did not smooth over the difficulties of its problems simply by making those direct assumptions. It will help to elucidate the nature of the matter in hand if here, too, we indicate the details of his method in one or two examples. The declared object of his method is, precisely, to provide an independent proof of the fact that between particular elements of a mathematical whole, for example, of a curve, there exists a relation of the original to the derived function. Now this proof cannot be effected in a direct manner because of the nature of the relation itself in this domain ; in the mathematical object this relation connects terms which are qualitative distinct, namely, curves with straight lines, linear dimensions and their functions with plane or surface dimensions and their functions, so that the required determination can only be taken as the mean between a greater and a less. Consequently, there spontaneously enters again the form of an increment with a plus and minus and the energetic ’developpons’ is here in place ; but we have already pointed out that here the increments have only an arithmetical, finite meaning. From the development of the condition that the required magnitude is greater than the one easily determinable limit and smaller than the other, it is then deduced that, e.g. the function of the ordinate is the derived, first function of the function of the area.
Lagrange’s exposition of the rectification of curves in which he starts from the principle of Archimedes is interesting because it provides an insight into the translation of the Archimedean method into the principle of modern analysis, thus enabling us to see into the inner, true meaning of the procedure which in the other method is carried out mechanically. The mode of procedure is necessarily analogous to the one just indicated. The principle of Archimedes, that the arc of a curve is greater than its chord and smaller than the sum of the two tangents drawn through the end points of the arc and contained between these points and the point of intersection of the tangents, gives no direct equation, but simply postulates an endless alternation between terms determined as too great or too small, the successive terms always being still too great or too small but within ever narrower limits of inaccuracy ; its translation into the modern analytical form, however, takes the form of finding an expression which is per se a simple fundamental equation. Now whereas the formalism of the infinitesimal directly presents us with the equation dz2 = dx2 + dy2, Lagrange’s exposition, starting from the basis indicated, demonstrates that the length of the arc is the original function to a derived function whose characteristic term is itself a function coming from the relation of a derived function to the original function of the ordinate.
Because in Archimedes’ method, as well as later in Kepler’s treatment of stereometric objects, the idea of the infinitesimal occurs, this has often been cited as an authority for the employment of this idea in the differential calculus, although what is peculiar and distinctive in it has not been brought specifically to notice. The infinitesimal signifies, strictly, the negation of quantum as quantum, that is, of a so-called finite expression, of the completed determinateness possessed by quantum as such. Similarly, in the subsequent celebrated methods of Valerius and Cavalieri, among others, which are based on the treatment of the relations of geometrical objects, the fundamental principle is that the quantum as such of the objects concerned, which are primarily considered only in their constituent relations, is for this purpose to be left out of account, the objects thus being taken as non-quantitative.
However, in these methods the affirmative aspect as such which is veiled by the merely negative determination fails to be recognised or brought to notice — that aspect namely which above presented itself abstractly, as the qualitative determinateness of quantity, and more precisely, as lying in the relation of powers ; and also, since this relation itself embraces a number of more precisely determined relations such as that of a power and the function of its development ; these also, in turn, are supposed to be based on and derived from the general and negative determination of the same infinitesimal. In the exposition of Lagrange just noticed, the specific affirmative aspect which is implied in Archimedes’ method of developing the problem is brought to notice with the result that the procedure which is burdened with an unlimited progression is given its proper limit. The greatness of the modern invention per se and its capacity to solve previously intractable problems and to treat in a simple manner those previously soluble, is to be ascribed solely to the discovery of the relation of the original to the so-called derived functions and of those parts of a mathematical whole which stand in such a relation.
What has been said may suffice to signalise that distinctive relation of magnitudes which is the subject matter of the particular kind of calculus under discussion. It was possible to confine our exposition to simple problems and the methods of solving them, it would neither have been expedient as regards the determination of the Notion, which determination is here our sole concern, not would it have lain in the author’s power to have reviewed the entire compass of the so-called application of the differential and integral calculus, and by reference of all the respective problems and their solutions to what we have demonstrated to be the principle of the calculus, to have carried out completely the induction that the application is based upon this principle. But sufficient evidence has been produced to show that just as each particular mode of calculation has as its subject matter a specific determinateness or relation of magnitude, such relation constituting addition, multiplication, the raising to powers and extraction of roots, and operations with logarithms and series, and so on, so too has the differential and integral calculus ; the subject matter proper to this calculus might be most appropriately named the relation between a power function and the function of its expansion or potentiation, because this is what is most readily suggested by an insight into the nature of the subject matter.
Logarithms, circular functions and series are of course also employed in the calculus, especially for the purpose of making expressions more amenable for the operations necessary for deriving the original function from the functions of expansion ; but they are only used in the same way that the other forms of calculation such as addition, etc., are also used in the calculus. The differential and integral calculus has, indeed, a more particular interest in common with the form of series namely, to determine those functions of expansion which in the series are called coefficients of the terms ; but whereas the calculus is concerned only with the relation of the original function to the coefficient of the first term of its expansion, the series aims at exhibiting in the form of a sum, groups of the terms arranged according to powers which have these coefficients. The infinite which is associated with infinite series, the indeterminate expression of the negative of quantum in general, has nothing in common with the affirmative determination belonging to the infinite of this calculus. Similarly, the infinitesimal in the shape of the increment, by means of which the expansion is given the form of a series, is only an external means for the expansion, and the sole meaning of its so-called infinity is to have no other meaning beyond its significance as such means ; the series, which in fact is not what is wanted, produces an excess, the elimination of which causes the unnecessary trouble. The method of Lagrange, who preferred to use the form of series again, is also burdened with this difficulty ; although it is through his method, in what is called the application, that what is truly characteristic of the calculus is brought to notice, for, without forcing the forms of dx, dy and so on, into the objects, it is directly demonstrated to which part of the object the determinateness of the derived function (function of expansion) belongs ; and thus it is evident that the matter in hand here is not the form of series.
[In the critique quoted above are to be found interesting views of a profound scholar in this science, Herr Spehr ; they are quoted from his Neue Prinzipien des Fluentenkalkuls, Brunswick, 1826, and concern a factor which has materially contributed to what is obscure and unscientific in the differential calculus and they agree with what we have said about the general character of the theory of this calculus. ’Purely arithmetical investigations,’ he says, ’admittedly those which have a primary bearing on the differential calculus, have not been separated from the differential calculus proper, and in fact, as with Lagrange, have even been taken to be the calculus itself whilst this latter was regarded as only the application of them. These arithmetical investigations include the rules of differentiation, the derivation of Taylor’s theorem, etc., and even the various methods of integration. But the case is quite the reverse, for it is precisely those applications which form the subject matter of the differential calculus proper, all those arithmetical developments and operations being presupposed by the calculus from analysis.’ We have shown how, with Lagrange, it is just the separation of the so-called application from the procedure of the general part which starts from series, which serves to bring to notice the characteristic subject matter of the differential calculus. It is strange, however, that the author, who realises that it is just these applications which form the subject matter of the differential calculus proper, should get involved in the formal metaphysics (adduced in that work) of continuous magnitude, becoming, flow, etc., and should want to add even fresh ballast to the old ; these determinations are formal, in that they are only general categories which do not indicate just what is the specific nature of the subject matter, this having to be learned and abstracted from the concrete theory, that is, the applications.]