why i don’t believe in manipulation theory of media
his fundamental proposition of the theory of gravitation in Princ. mathem. philosopfuae naturalis lib. /., Sect. II., Prop. I., and cf. Schubert’s1 * 3 Astronomy — first Ed., Vol. III.,§ 20 — , where it is admitted that things are not exactly as Newton assumes, that is, at that point which is the nerve of the proof.)
It will be impossible to deny that in this sphere much has been accepted as proof— chiefly veiled under the kindly mist of the infinitesimally small — on no other ground than that the result was always already known beforehand, and the proof, which was arranged in such a manner as to produce the result, at least effected the illusion of a framework of proof, — which illusion was preferred to mere belief or empirical knowledge. But I do not hesitate to regard this method as no better than demonstrational jugglery and counterfeiting; and I include even some of Newton’s demonstrations, and especially such as belong to those just mentioned, for which Newton has been extolled to the skies and above Kepler, because what Kepler had discovered empirically he demonstrated mathematically.
The useless framework of such proofs was erected in order to prove physical laws. But mathematics is altogether unable to prove the magnitudinal determinations of physics in so far as they are laws based upon the qualitative nature of the moments, for this simple reason that this science is not philosophy and does not start from the concept, so that the qualitative element (in so far as it is not taken lemmatically from experience) lies outside its sphere. The assertion of the honour of mathematics, which demands the strict proof of all its propositions, often allowed it to forget its limit; thus it seemed against its honour to acknowledge simply experience as source and sole proof of empirical propositions. Thought at a later period achieved a more instructed view of this matter; and, until it clearly understands the difference between what is mathematically demonstrable and what can only come from another source, between the things which are terms of an analytical development and those which are physical existents, scientific method cannot achieve a strict and pure attitude. — But justice will no doubt be done to that framework
1 Schubert, Friedrich -Theodor von, 1 758-1895, Director of the Observatory
at S. Petersburg: Lthrbuch dir thmretischtn Astronomit, 1798; Populate Astronomii,
of Newtonian proof as it was to another baseless Newtonian structure of optical experiments and conclusions connected with them. Applied mathematics still seethes with a similar brew of experience and reflection ; but long ago one part after another of that optics began to be ignored by science in practice (with this inconsistency — that all the rest, though in con¬ tradiction, was allowed to stand) ; and, similarly, in practice some of these sophistical proofs have already been forgotten or replaced by others.
The Purpose of the Differential Calculus deduced from its Application
In the previous Observation we considered, partly the con¬ ceptual determinateness of the infinitesimally small, which is used in the differential calculus, and partly the reason for its introduction into this calculus ; both determinations are abstract and in themselves therefore easy. Their so-called application, however, offers greater difficulties but also a more interesting aspect: the elements of this concrete side are to be the object of this Observation. — The whole method of the differential calculus is complete in the proposition that
d (*“) = n x*-1 d x, or~^* — — = P , that is, that it is equal
to the coefficient of the first term of the binomial x + dx, x + t, if the latter is developed according to the powers of dx or i. No need to learn anything further : the derivation of the next forms, the differential of a product, an exponential magnitude and so on, results mechanically; and in a short time, perhaps a mere half-hour — for the deduction of the differential also gives us the reverse, namely, the deduction of the original func¬ tion, or integration, from these former — the whole theory may be had by heart. The only delay is due to the effort to understand and make it intelligible that that other part — the omission of the remaining terms of the series which arises, apart from the first terms — is valid, after the first part of the task, the deduction of the coefficient, was effected so easily by analytical, that is, by purely arithmetical means, through the development of the
first, here’s what i think manipulation theory of media looks like
function of the variable magnitude, after an increment had given it the form of a binomial. And if it were the case that we needed only the coefficient, then, as was said, once it was determined the entire theoretical part would be done with in less than half an hour, and the omission of further terms of the series would cause no difficulty if only because they would not even come into question as terms of the series ; being second, third, or other functions, they are determined with the determination of the first, and now are quite beside the issue.
And first we may observe that probably the method of the differential calculus shows on the face of it that it was neither invented nor constructed as an end in itself. Not only was it not founded for its own sake as a new kind of analytical process, but the arbitrary neglect of terms resulting from the development of a function is contrary to every mathematical principle; — arbitrary, because it is assumed that the whole of this development entirely belongs to the matter in hand — the matter being looked at as the difference of the developed function of a variable magnitude (after it has been given the form of a binomial) from the original function. The need for such a mode of procedure and its internal lack of justification immediately point to the fact that its origin and basis must be elsewhere. It happens in other sciences, too, that that which is placed first, as being elementary, and which is the source from which are ostensibly derived the propositions of the science, is not self-evident, and that it eventually appears that its reason and foundation lie rather in what follows. In the history of the differential calculus the course of things makes it plain that, especially in the various so-called tangential methods, the matter began in special artifices. The method of procedure, later becoming extended to further objects, reached conscious¬ ness and was cast into abstract formulae, which it was then attempted to raise to the rank of first principles.
We have shown that the qualitative quantity-determinateness of entities, which, primarily, are related to one another as Qpanta, is the conceptual determinateness of the so-called infinitesimally small; from this there started the empirical investigation which attempted to demonstrate this conceptual determinateness in the descriptions or definitions of the in¬ finitesimally small, in so far as it is taken as infinitesimal
difference or something similar. — This was done only in the interest of abstract conceptual determinateness as such; it might further be asked, what is the nature of the transition thence to mathematical formation and application? To this end the theoretic part (the conceptual determinateness) must further be examined, and this will prove not wholly unfruitful in itself; next we must consider its relation to Application; and with both we must demonstrate, as far as here is possible, that the general conclusions are adequate to the end of the differential calculus and to the manner in which it brings that end about.
It must first be remembered that the mathematical form of the conceptual determinateness which is under discussion has already been mentioned in passing. The qualitative deter- minateness of the quantitative is first and generally stated in Quantitative Ratio, but already in the demonstration of the various so-called rules of arithmetic (see Observation on this subject) it was said that it was in the Ratio of Powers (to be considered later in its proper place) that Number is posited by the equation of Unit and Amount (the moments of its concept) as returned to itself, and thus acquires in itself the moment of Infinity or Being-for-Self, that is, of self-deter¬ minedness. Thus, the express qualitative magnitudinal deter¬ minateness essentially (as has also been mentioned already) refers to determinations of powers ; and since it is the specific characteristic of the differential calculus that it operates with qualitative forms of magnitude, its peculiar object in mathe¬ matics must be the treatment of forms of powers ; and all the problems, with their solutions, for which the differential calculus is used, show that all the interest lies only in the treatment of determinations of powers as such.
This foundation is important and immediately puts in the forefront something determinate instead of the merely formal categories of variable, continuous, or infinite magnitudes and the like, or of functions in general ; but for all that it is still too general; other operations too are concerned here; the elevation to a power and the extraction of a root, the treat¬ ment of exponential magnitudes and of logarithms and series and the equations of higher orders, all are interested in and operate upon ratios only which are based upon powers. No
doubt they must between them constitute a system of treatment of powers; which, however, of the various ratios wherein determinations of powers can be posited is the one which is the proper object and interest of the differential calculus, this can only be elicited from itself, that is, from its so-called applications. These in fact are the thing itself, the actual procedure in the mathematical solution of a certain group of problems; and this procedure was earlier than the theory or general part, and has been called application only with refer¬ ence to the later created theory whose aim was to set up the general method of procedure and also to produce its first principles, that is, its justification. It has been shown in the previous Observation how idle an attempt it has been to find principles for the former manner of comprehending the pro¬ cedure, — principles which really solved the contradiction which there was found, instead of excusing it or hiding it behind the insignificance of that which was requisite to the mathematical procedure (which here meant that which was to be omitted), or behind the possibility (which comes to the same thing) of infinite or arbitrary approximation, and the like. If the general part of the procedure were to be abstracted, in a manner different from that which has hitherto been followed, from that real part of mathematics which is called differential calculus, then these principles and all the trouble taken over them would prove superfluous, inasmuch as they reveal in themselves a distortion and an abiding contradiction.
If we search out this peculiarity by simply gathering what we find in this part of mathematics, we discover as its object : — (a) Equations in which any number of magnitudes (we may, however, here confine ourselves to two) are combined into one determinate whole in such a manner that, firstly, they have their determinateness in empirical magnitudes which are their fixed limits, and, moreover, in the particular kind of union with them and with one another, which, indeed, is the case with equations generally; but, since there is only one equation for both magnitudes (or several equations for several magnitudes, but always fewer than the magnitudes), these equations belong to the class of indeterminate equations ; — and, secondly, that one aspect of them (the determinateness of these magnitudes here being what it is) is that they are, or at least
one of them is, present in the equation in a higher power than the first.
Certain observations may be made about this; and first, that according to the first determination mentioned these magnitudes are wholly of the character of such variable mag¬ nitudes as occur in the problems of indeterminate analysis. Their value is indeterminate, but is so in such a manner that when the one gets a perfectly determinate value (a numerical value) from without, then the other too is determined: one is a function of the other. The categories of variable magnitudes, functions, and the like, are, therefore, merely formal for that specific determinateness of magnitude with which we are dealing here, as was said above; for they are of a generality not yet containing that specific factor which is the one aim of the differential calculus; nor can that factor be explained from them analytically. In themselves they are simple, unim¬ portant, easy determinations, which are made difficult only when what they do not properly contain is put into them in order to be then deduced, I mean the specific determination of the differential calculus. — With regard to the so-called constant, we may remark that it exists first as an indifferent empirical magnitude, determining the variable magnitudes only with respect to their empirical Quantum, as Limit of their minimum and maximum; while the nature of the connexion between the constants and the variable magnitudes is itself one of the moments of the nature of that special function which these magnitudes are. But, conversely, the constants also are functions; for instance, in so far as a straight line has the
1 the media is controlled by powerful humans
meaning that it is the parameter of a parabola, then its meaning
is this, that it is the function - ; and, generally, in the develop¬
ment of the binomial, 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 their products, in pairs, and so on : thus the constants here are just functions of the roots; where, in the integral calculus, the constant is determined from the given formula, it is in this respect treated as a function of the latter. We shall elsewhere consider these coefficients, in another determination, as functions, whose meaning in the concrete is the only matter of interest.
The peculiarity, however, by which the consideration of the variable magnitudes is distinguished, in the differential calcu¬ lus, from its nature in the indeterminate problems, must be attributed to what has been mentioned, namely that at least one, and possibly every one, of these magnitudes is in a higher power than the first; and here again it is indifferent whether all of them are of the same higher power, or of unequal powers ; the specific indeterminateness which they here have lies solely in the fact that they are functions of one another standing in such a ratio of powers. This gives a qualitative determination to the variation of the variable magnitudes : it is thus continuous, and this continuousness (which in itself is merely the formal category of an identity in general, a determinateness persisting unchanged in variation) here has its determinate meaning, and that only in the ratio of powers, which is supposed to have no Quantum for exponent and to constitute the non-quanti- tative and permanent determinateness of the ratio of variable magnitudes. On the other hand we may object to another formalism, that the first power is power only in relation to higher power: for itself, * is only any indeterminate Quantum. There is thus no meaning in differentiating in themselves the equations y — a x -f- b (the equation of the straight line) or s = ct (the equation of plain uniform velocity) ; if the
equation y — ax, or y — a x + b, becomes a — or s — ct
becomes — = c, then, equally, a — — is the determination of at x
the tangents, or - = c the determination of plain velocity. dy
The latter, as -f-, is exhibited in connexion with what is dx
asserted to be the development of uniformly accelerated motion ; but it has been remarked above that it is an empty assumption, based upon the routine of method alone, that a moment of simple, merely uniform velocity (that is, velocity not deter¬ mined by the higher power of one of the moments of motion), has a place in the system of such motion. The method proceeds from the idea of the increment to which the variable magnitude is subject; naturally, therefore, only a magnitude which is a function of a first power can be subject to an increment : when
now, in order to discover the differential, the difference be¬ tween the given equation and that which has thereby arisen, has to be found, the empty nature of this operation manifests itself; for, as we remarked, after as well as before it the equation is the same for the so-called increments as for the variable magnitudes themselves.
2 they wield it instrumentally for their goals external to the medium
(/?) What has been said determines the nature of the equation which is to be treated ; we must now indicate upon what point of interest its treatment is directed. This consideration can furnish only known results, such (with respect to form) as are especially to be found in Lagrange’s conception; but I have, of course, made this exposition wholly elementary in order to remove all heterogeneous determinations involved in it. — The basis of the treatment of an equation of this kind is seen to be this, that the power within itself is taken as a ratio or system of determinations of ratios. We stated above that Power is Number in so far as it has reached that stage where its variation is determined by itself, and its moments, Unit and Amount, are identical; — which, as was shown above, is found perfectly in the square, and more formally (which here makes no difference) in the higher powers. Now power is number (though the expression “magnitude” be preferred as more general, yet in itself it always is number), and thus is a mul¬ titude, and can be represented as a sum: it can, therefore, be divided within itself into any multitude of numbers, which relatively to one another and to their sum are without any determination except that together they are equal to the sum. But the power can also be divided into a sum of such differ¬ ences as are determined by the form of the power. If the power is taken as sum, then its radical number or root too is taken as a sum, and, as regards multiplicity of division, is arbitrary, — which multiplicity, however, is the indifferent and empirical Quantitative. The sum, in which shape the root is supposed to exist, reduced to its simple determinateness, that is, its true universality, is the binomial ; and any further multiplication of the terms is a pure repetition of the same determination, and, therefore, an empty process.1 What matters is the determinate-
1 It is only a part of the formalism of that universality to which analysis per¬ force lays claim, when (a 4- 4) * is not taken for the development of the power, and (a+ i + c + d+ ...)’• is substituted. This is often done elsewhere too; and this
ness of the terms (which thus is qualitative) which results from the raising of the root, taken as sum, to a power, and this determinateness lies entirely within that change which this process is. These terms, thus, are entirely functions of power and of raising to a power. Now this representation of number as sum of a multitude of such terms, which are functions of this process, and further the eagerness to find the form of such functions and also this sum from the multitude of such terms, this it is (the discovery being entirely dependent on that form) which constitutes, as is well known, the special doctrine of series. But here it is important to distinguish the further point of interest, namely, the relation of the basic magnitude itself (the determinateness of which, in so far as it is a complex, that is, in this instance, an equation, therefore includes a power) to the functions of its potentiation. This relation (apart altogether from the above-mentioned interest in the sum) will show itself to be the only truly scientific point of view by which the differential calculus is guided.
But first another determination must be added to what has been said; or rather, a determination implied in it must be removed. It was said that the variable magnitude, in whose determination power is an element, is looked upon as a sum within itself, as being, in fact, a system of terms, in so far as these are functions of potentiation, and that thus the root too was considered as a sum, and, in its simply determinate form, as a binomial, namely
This presentation started from Sum as such upon the process of developing Power, that is, upon the process of achieving its potentiational functions; here, however, the aim is not a sum as such, nor the series which arises from it: the only element which must be taken up from Sum is Relation. Relation of magnitudes as such is on the one hand all that remains after abstraction has been made from the plus of a sum as such, and is all that is needed on the other hand in
form, «o to speak, should be taken for an affectation of the appearance of univer¬ sality, for the matter is exhausted in the binomial; by the development of this the law is found, and it is the law which is the true universality, and not external and empty repetition of the law, which is all that this «+ * + e+d+ . . . effects.
order that the functions of development of Power may be found. But such a Relation is determined already by the fact that the object here is an equation {jT—af), that is, a complex of several (variable) magnitudes which contains their determination of powers. In this complex each of these mag¬ nitudes is posited as just being in relation with the other, with the meaning, as one might say, of a plus in itself, — as a function of the other magnitudes; their characteristic (that of being functions of one another) gives them this determination of a plus, which for that very reason is quite indeterminate and is not an addition, increment, or the like. This abstract point of view could, however, be neglected ; we may simply remain at that point where, the variable magnitudes being given in the equation as functions of one another in such a manner that this determinateness contains a relation of powers, the functions of potentiation too are now compared one with the other; — which second functions have no determination what¬ ever except that which comes from potentiation. So far we can assert that it is optional, or possible, to transpose an equation from the powers of its variable magnitudes to a relation of its functions of development ; and the utilitarian quality of such a transformation must be indicated by some further purpose, advantage, or use; the transformation has been made only because of its usefulness. Above we started from the presen¬ tation of these determinations of potentiation, and experimented on a magnitude which was stated to be a sum and, therefore was to be assumed as being differentiated within itself; but this was only done, partly to indicate the nature of such functions, and partly because this implies the mode by which they are found.
We have thus reached the ordinary analytical development, which for the purposes of the differential calculus is taken in this manner, that an increment, dx or i, is given to the variable magnitude, and then the power of the binomial is explained by means of the series of terms belonging to it. The so-called increment, however, is intended to be not a Quantum but a form, whose only value is that it assists the development; what is wanted (and this is admitted, and most explicitly by Euler and Lagrange) in the above-mentioned idea of Limit, is the resulting determinations of powers of the variable
magnitudes, the so-called coefficients of the increment (as is admitted) and its powers, according to which the series is ordered, and to which the different coefficients belong. To this we might also add that an increment (without Quantum) is only assumed for the sake of development, and that, therefore, it would be most convenient to take 1 (the One) for that purpose; for in the development this only occurs as factor; so that the factor One fulfils the purpose, which is, that the increment is not to imply the positing of any quantitative determinateness or change ; dx, on the other hand, is infected with the false idea of a quantitative difference, and other symbols, like t, with the show, useless here, of universality, so that they always have the appearance and pretension of a Quantum and its powers ; which pretension then involves the trouble that they must nevertheless be taken away and kept away. In order to preserve the form of a series developed on the principle of powers, the denominations of exponents (being indices) might equally well be placed behind a One. In any case abstraction must be made from the series and from the determination of coefficients according to their place in the series : the relation between all is the same; the second function is derived from the first in just the same manner as this from the original, and for the one which is counted second, the first, and derivative, function, is counted original. The essential point of interest is not, however, the series, but solely the determination of powers resulting from the development in relation to the magnitude which, to them, is immediate. They are not, therefore, deter¬ mined as being coefficients of the first term of the development, since one term is designated as first in relation to others follow¬ ing it in the series, and such a power as the power of an incre¬ ment, together with the series, is here out of place ; therefore, the plain expression of derivative function of a power or, as was said above, function of potentiation of a magnitude, would be preferable, — the knowledge being presupposed of the manner in which the derivation is taken as a development included within a power.
The pure mathematical beginning in this part of the analysis is just the discovery of the function determined by the develop¬ ment of powers; it is a further question, what is to be done with the ratio thus obtained, where it is to be applied and usedf
3 the contents disseminated through the media shape the worldview of “the people”
or, in fact, for what purpose such functions are looked for. It is the process of discovering proportions in concrete objects which may be referred back to those abstract analytic pro¬ portions, that has given the differential calculus its great interest.
But as regards applicability, the immediate outcome of the nature of the matter — merely in virtue of the form which we have shown the moments of powers to possess, and no con¬ clusion being yet drawn from any instances of application — is as follows. The development of powers whence result the functions of their potentiation contains (abstraction being made from closer determination) the reduction of magnitude to the next lowest power. Thus it is that this operation is applicable to such objects as also contain this distinction of determinations of powers. If now we consider spatial determinateness, we find that it contains the three dimensions which we may call the concrete, in order to distinguish them from the abstract distinc¬ tions of height, length, and breadth, — namely, line, surface, and solid space; and, when they are taken in their simplest forms and with reference to self-determination and, therefore, to analytic dimensions, we arrive at the straight line, the plane surface and surface taken as square, and the cube. The straight line has an empirical Quantum, but with the plane we reach the qualitative element — determination of power; we may neglect closer modifications, — the reflection, for instance, that this also happens to plane curves ; for here we are only dealing with the distinction in general. Herewith also the need arises to pass from a higher to a lower determination of power, and conversely, — the attempt being for instance to derive linear determinations from given equations of plane and so on, or the other way about. — In motion, further, the magnitudinal proportion of the space passed through with its elapsed time must be considered, and motion manifests itself in various determinations, as simply uniform, as uniformly accelerated, and as alternately accelerated uniformly and retarded uni¬ formly, returning upon itself; these different kinds of motion are expressed according to the magnitudinal proportion of their moments, space and time; and thus there result for it equations having different determinations of powers; and, in so far as it may be necessary to determine one kind of motion or of
spatial magnitude to which one kind is tied, from another kind of these, the operation also involves the transition of one function of power to another, either higher or lower. — The examples of these two objects should suffice for the purpose for which they are cited.
The appearance of contingency which the differential cal¬ culus presents in its applications would certainly be simplified if the nature of the spheres where the application can take place, and the peculiar need for and condition of this applica¬ tion, were clearly understood. But now it is necessary further to know, within these spheres, between what parts of the objects of the mathematical problem such a relation takes place as is posited peculiarly by the differential calculus. And first it must be remarked that two kinds of relation are to be observed. The operation of the depotentiation of any equation (the equation considered according to the derivative functions of its variable magnitudes) gives a result which in itself really no longer is an equation but a proportion: this proportion is the object of the differential calculus proper. And precisely in this fact we are also presented with a second proportion, that between the higher determination of power (the original equation) itself and the lower (the derivative). This second proportion we must here provisionally neglect: it will prove to be the peculiar object of the integral calculus.
We will then consider the first relation, and, for the deter¬ mination of the moment (which must be taken from the so- called application and contains the interesting part of the operation) we will take the simplest example, curves such as are determined by an equation of the second degree. The relation of the coordinates is, of course, given immediately through the equation in a determination of powers. From the fundamental determination there follow determinations of other straight lines connected with the coordinates — tangents, subtangents, normals, and so on. The equations, however, between these lines and the coordinates are linear equations; and the wholes, of which these lines are determined to be parts, are right-angled triangles of straight lines. Now the transition from the fundamental equation, which contains the determination of the powers, to these linear equations, con¬ tains the above transition from the original function (that is.
the function which is an equation) to the derivative function (which is a relation — a relation subsisting between certain lines contained in the curve). What must now be discovered is the connexion between the relation of these lines and the equation of the curve.
It is not without interest to introduce this much of historicity, as to remark that the first discoverers can indicate their dis¬ covery only in a wholly empirical manner, without being able to render an account of the operation, which has remained quite external. In this regard I content myself with a reference to Barrow, who was Newton’s master. In his led. Opt. et Geom., in which he treats problems of higher geometry according to the method of the indivisibles (which, in the first place, differs from the characteristic feature of the differential calculus), he also indicates his mode of determining the tangents, “because his friends had urged him” {led. X.). The nature of this indication must be read in his own words if we would properly understand how this method is given quite as an external rule — in the same style in which formerly in arithmetical school¬ books the “rule of three,” or, better still, the so-called “test by casting out the nines,” was enunciated. He enumerates the minute lines, which later were called the increments in the characteristic triangle of a curve, and then gives the instruction — which is a mere rule — to cast off as superfluous the terms which appear when the equations are developed as powers of these increments or products ( etenim isti termini nihilum valebant) ; adding that the terms which contain only magnitudes determined by the original equation must also be cast off ( — which means that the original equation is sub¬ tracted afterwards from the equation formed with the incre¬ ments), and that the ordinates themselves must be substituted for the increment of the ordinates, and the subtangents for the increment of the abscissae. The method could be set forth (if one may say so) in no more school-masterly manner; — the second substitution is the assumption of the proportionality of the increments of ordinates and abscissae to the ordinates and subtangents, which is the basis of the procedure for deter¬ mining tangents in the ordinary differential method : in Barrow’s rule this assumption appears in all its naive naked¬ ness. A simple way of determining subtangents had been
found: Roberval’s1 and Fermat’s methods come to the same issue; — the method for finding maximal and minimal values, from which the latter started, is based on the same foundation and the same procedure. It was a mathematical mania of that period to discover so-called methods, that is, rules of this kind, and to make a mystery of them ; which was not only easy, but also, in one respect, necessary, and for the same reason for which it was easy, — namely that the inventors had found only an empirical and external rule and no method, that is, nothing derived from acknowledged principles. Such so-called methods Leibniz absorbed from his period, and Newton from it and, immediately, from his master: the generalization implied in their form and applicability opened new roads to the Sciences, but also brought the need of forcing the method out of the shape of merely external rules, and attempted to give it the justification which it demanded.
If now the method be more closely analysed, the true process is this. First the determinations of powers (powers, of course, of variable magnitudes) contained in the equation, are reduced to their first derivatives. But this changes the value of the members of the equation : hence no equation remains, and a proportion has arisen which subsists between the first derivative of one variable magnitude and the first derivative of the other; for fix =jp we have p : 2 y, or for 2 a x — xt —y% we have a — x :y, which later used to be called the proportion dy dx
on the other hand, wholly dependent on it and wholly derived from it (derived in the process described above in accordance with a bare rule), is linear, certain lines here being proportional one to the other; p : ay or a—x:y are themselves relations of straight lines of the curve, or of coordinates or parameters ; but so far knowing this we know nothing. The aim is to know, with regard to other lines which occur in connexion with the curve, that a certain one of these relations holds between them, — that is, to identify two relations. — Thus, secondly, the question arises, which are the straight lines determined by the nature of the curve which are thus related? — But this was known already — namely, that a relation thus reached is the
this is a very compelling theory, especially at this point in history. it seems like every billionaire is buying their own media platform which seemed so revolutionary just a few years ago. if only we could go back…
relation between ordinates and subtangents. The ancients had found this by an ingenious geometrical method ; what modem inventors discovered was the empirical mode of arranging the equation of a curve in such a manner that the first relation should result, of which it is already known that it is equal to a relation which contains the line (here the subtangent) which it is desired to determine. Now, in part, this arrangement of the equation has been taken methodically and made methodi¬ cally (Differentiation), — and, in part, the imaginary increments of the coordinates and the imaginary characteristic triangle (which is formed of these and of a similar increment of the tangents) have been invented, — simply in order that the pro¬ portionality between the relation found by the depotentiation of the equation and the relation between ordinates and sub¬ tangents should not be represented as taken up empirically from old acquaintance, but as a demonstrated truth. And yet old acquaintance proves itself, generally and most unmistak¬ ably in the form of rules (such as we quoted), as the only occasion and justification (where needed) of the assumption of the characteristic triangle and of that proportionality.
It was Lagrange who rejected this pretence and followed the true scientific course: thanks to his method we know the real point at issue, for it consists in a separation of the two transitions which must be made in order that the problem may be solved, and in a separate treatment and proof of each of these sides. — In our more detailed explanation of the pro¬ cedure we will retain as example the elementary problem of finding the subtangents. — Now one part of this solution, the theoretical or general part, which is the finding of the first function from the given equation of curves, is regulated by itself: this part gives us a linear relation, a relation, that is, of straight lines, which occur in the system of curve-deter¬ mination. The other part of the solution now is the finding of those lines in the curve which are in this relation. This is done directly ( Theorie des Fonct. Anal. II. P. II. Chap.), that is, without the characteristic triangle, which means that no assumption is made of infinitely small arcs, ordinates, or abscissae, nor are the determinations of dy and d x (that is, of being sides of this relation) attributed to them, which would n^:an immediately that it was equated with the ordinate and
subtangent. A line (and also a point) has its determination in so far as it constitutes the side of a triangle, and the deter¬ mination of a point too lies only in this. This (as may be mentioned in passing) is the fundamental proposition of analytical geometry, which introduces the coordinates as (what is the same thing) in mechanics it does the parallelogram of forces, — which for that very reason does not require all the efforts that are made to find a proof. — The subtangent is now made the third side of a triangle, the other sides of which are the ordinate and the relative tangent. The latter is a straight line, and, therefore, its equation is p — aq, (the addition of b adds nothing to the determination and is made only for the sake of the fetish of universality) ; — the determination of
the ratio - falls within a, the coefficient of q, which in turn
is the first derivative of the equation, but need be considered
only as a — -, — being, as has been said, the essential deter-
mination of the straight line which is applied as tangent to the curve. Now, further, the first derivative of the curve- equation is taken, and, therefore, it is also the determination of a straight line ; further, it is assumed that the coordinate p of the first straight line, and y, the ordinate of the curve, are identical, and, therefore, that the point where that first straight line (which is taken to be a tangent) touches the curve is also the beginning of the straight line which is determined by the first function of the curve : the task, therefore, is to show that this second straight line coincides with the first and, therefore, is a tangent; which may be thus algebraically expressed: — since y —fx and p = F q, and it is assumed that y = p and hence that fx—Fq, therefore f'x — F'q. In order to prove that the straight line which is applied as tangent and the straight line of the equation which is determined as being its first function coincide, and, therefore, that the latter is a tangent, recourse is had to the increment t of the abscissa and to the increment of the ordinate which is determined during the development of the function. Thus here too that ill-famed increment is introduced ; but its introduction for this purpose, and the development of the function under its guidance, must carefully be distinguished from the use (mentioned above) ,of
the increment in the discovery of the differential equation and in the characteristic triangle. The use here made is justified and necessary; it falls within the scope of geometry, for it is part of the geometrical determination of a tangent as such that between it and the curve with which it has a point in common no other straight line can pass and pass also through that point. In this determination the quality of tangent or not-tangent is reduced to a magnitudinal difference : that line is the tangent of which simply greater smallness with respect to the essential determination is predicated. There is no empirical element whatever in this smallness, which appears to be only relative, — nothing, that is, which depends on a Quantum as such; it is posited as qualitative by the nature of the formula when the difference of the moment on which the magnitude which is to be compared depends, is a difference of powers; this difference is reduced to i and i*, and i (which after all must signify a number) must then be imagined as a fraction, and thus il in itself is smaller than i; so that the idea of an arbitrary magnitude as which i might be taken is here superfluous and even out of place. And for this very reason the demonstration of a greater smallness has nothing to do with an infinitesimally small quantity, the introduction of which is thus here by no means necessary.
I will now mention— if only for its beauty and for its fame, mostly forgotten now, but well deserved — the tangential method of Descartes; it moreover has a bearing on the nature of equations, on which an observation remains to be made. Descartes unfolds this independent method, where the linear determination required is discovered from the same derivative function, in his geometry, which has proved so fruitful in other respects too (liv. II. p. 357 sqq. Oeuvres Compl. ei. Cousin Tom. V.) ; for in it he teaches the great basis of the nature of equations and their geometrical construction, and of the Analysis of Geometry, the sense of which he had thereby so greatly ex¬ tended. With him the form of the problem was this, to draw straight lines at right angles to given points of a curve, by which method the subtangent (and so forth) is determined; and it is easy to understand the satisfaction which he expresses there at his discovery, which concerned an object of general scientific interest at that period, and is purely geometrical and
thereby stands high above the mere methods of rule (mentioned above) used by his rivals: “ J'ose dire que c'est ceci le probleme le plus utile et le plus general, non seulerrunt que je sache, mais mime que foie jamais disire de savoir en giometrie." — He bases the solution upon the analytic equation of the right-angled triangle which is formed by (1) the ordinate of the point of the curve to which the straight line demanded in the problem is to be perpendicular, by (2) this straight line itself (the normal), and by (3) that part of the axis which is cut off by the ordinate and the normal (the sub-normal). Now the equation of a curve is known, and from this equation the value of ordinate or abscissa is substituted into this equation of the triangle. Thus an equa¬ tion of the second degree results ; and Descartes shows also how curves whose equations contain higher degrees are reduced to this. In this equation only one of the variable magnitudes occurs, either as square or as first power ; — a quadratic equation which at first appears as a so-called impure equation. On this Descartes reflects as follows : — If the point taken in the curve is imagined as the point of intersection of this and of a circle, then this circle will intersect the curve at another point, so that for the two unequal x’s which will thus arise there will be two equations with the same constants and having the same form; — or else there will be only one equation with unequal values of x. But the equation is one only for the one triangle, in which the hypotenuse is perpendicular to the curve (or is normal), — which is imagined in this way, that the two points of intersection of the curve and the circle are allowed to coin¬ cide, so that the curve is allowed to touch the circle. But then also the fact that the x or y of the quadratic equation has unequal roots, disappears. But in a quadratic equation of two equal roots, the coefficient of the member contained by the unknown in the first power is twice the single root ; and from this there results an equation by which the required deter¬ minations are discovered. This method must be considered the brilliant device of a true analytic mind, compared with which the arbitrarily assumed proportionality of subtangent and ordinate, together with the so-called increments of abscissa and ordinate (supposed to be infinitely small), is vastly inferior.
it genuinely is not exactly wrong, in my eyes, but i think it’s incomplete, and this incompleteness means if you just believe these 3 things, you act differently, and i fear that the difference in action, rather than challenging power, plays into power
The final equation reached in this manner, in which the coefficient of the second member of the quadratic equation
is equated with the double root or unknown, is the same as is found by the method of the differential calculus. When #* — ax — b — o is differentiated, there results the new equation sx — a — o; or, again, 3 x* — p = o results from ** — p x -r- q = o. And here we may observe that it is by no means self-evident that such a derivative equation is also correct. We have already considered the fact that in an equation with two variable magnitudes which never lose their quality of being unknown magnitudes just because they are variable, only a proportion results; and this for the simple reason indicated, that, when the functions of the potentiation are substituted for the powers themselves, the value of the two members of the equation is altered, and it remains as yet unknown whether an equation subsists between them with
their values thus altered. The equation -j- — P expresses
nothing except that P is a proportion, and no other real
meaning can be ascribed to And also it is still not known
of this proportion = P, to what other proportion it is equal; it is only this equation, or proportionality, which gives value and meaning to the proportion. — It was mentioned that this meaning — and this was what was called application — was introduced empirically and from without; and, similarly, as the equations here under discussion are derived by differentia¬ tion, we must know from some other source whether they have equal roots, in order to know whether the equation reached remains correct. But this fact is not expressly brought to notice in the manuals; although it must be admitted that it is got out of the way when an equation with an unknown, reduced to nought, is straightway equated with y, whereupon
tion. It is true that the functional calculus is supposed to deal with functions of potentiation, and the differential calculus with differentials ; but it by no means follows immediately that magnitudes whose differentials or functions of potentiation are taken, are themselves only to be functions of other magnitudes. In any case, in the theoretic part, that is, where the instruc¬ tion is given to derive the differential (or the functions of
potentiation), no thought is yet given to the intention that the magnitudes, of which the treatment according to such a derivation is there taught, are themselves to be functions of other magnitudes.
With regard to the omission of the constant in the process of differentiation, this further observation may be made, that differentiation here means that the constant is indifferent for the determination of the roots when they arc equal, the deter¬ mination being exhausted by the coefficient of the second member of the equation. In the example quoted from Descartes the constant is itself the square of the roots, so that it can be determined from the constant as well as from the coefficients ; for, generally, like the coefficients, it is a function of the roots of the equation. In the ordinary exposition the omission of the so-called constants (which are connected with the other members only by plus or minus) is brought about by the mere mechanism of the method, — when, in order that the differential of a composite expression may be found, only the variable magnitudes receive an increment, and the expression thus formulated is subtracted from the original expression. The meaning of the constants and of their omission, the question how far they are themselves functions and serve or do not serve in this determination, finds no expression here.
In connexion with the omission of constants, an observation may be made about the names of Differentiation and Inte¬ gration, similar to the observation which was made above about the expressions Finite and Infinite, — namely, that their deter¬ mination contains the opposite of what is denoted by the terms. To differentiate denotes that differences are posited, whereas differentiation in fact reduces an equation to lesser dimensions, and the omission of the constant removes one moment of the determinateness ; for, as we remarked, the roots of the variable magnitude were placed upon an equality, and thus the dif¬ ference between them was cancelled. And in integration the constant is supposed to be reintroduced; and though by this process the equation is integrated, it is so in this sense, that the difference of the roots, which had just been cancelled, is reconstructed, so that the equalization is differentiated once more. — The ordinary expression adds its share in obscuring the essential nature of the matter and in setting everything
that is, if we believe humans ultimately control the media, then obviously the question is how could we get good manipulators to accrue power so good pqtriots would be in control
in a point of view which is subordinate and even alien to the main issue : I mean the point of view of the infinitely small difference, the increment, and the like, and also of the bare difference generally between the given and the derivative function, no designation being made of the specific, namely the qualitative, difference.
Mechanics is another important field where the differential calculus is applied; mention has already been made inciden¬ tally of the different functions of powers that result from the elementary equations of its object, which is motion, and of their significance. I will admit these directly here. The equation (that is, the mathematical expression) for motion which is s
simply uniform, c = - or s — c t, where the spaces passed
through are proportionate to the times elapsed according to an empirical unit c (the magnitude of velocity), offers no meaning for differentiation: the coefficient c is already fully determined and known, and no further development of powers can take place. — The analysis of s = a P, the equation of the motion of a falling body, has already been recorded; — the first
member of the analysis ^ = 2 a t is translated into language
(or into fact, as the case may be) when it is postulated that a member of a sum (an idea which we banished long ago) must be one part of this motion ; and that, further, this part must be added to the force of inertia (a merely uniform velocity) in such a manner that the motion is uniform in infinitesimally small parts of time, and not uniform in finite parts of time, that is, in those which actually exist. It is true that f s = 2 at; and the meaning of a and t is known already, together with the fact that this suffices to posit the deter-
mination of the uniform velocity of a motion ; for since a =
then 2 at — — universally ; but having this we are no wiser
at all, and only the deceitful assumption that 2 a I is part of the motion regarded as a sum gives the deceitful appearance of a physical proposition. The factor itself (a, the empirical unit, a Quantum as such) is ascribed to gravity; but if the category of force of gravity is employed at all, it should rather be said
that the whole s — aP is the effect, or, better, the law, of gravity. — Similar to this is the proposition derived from is
if we believe that the media can be used instrumentally for goals exterior to the medium, then good manipulators are even more important, because if we can control the media, then we can exert influence on other parts of the world
— — 2at, which enunciates that if gravity ceased to act, the
body, moving at the velocity reached at the end of its fall, would describe twice the space it has already passed through in a time equal to that occupied by its fall. — This contains a metaphysic which in itself is unsound : the end of the fall, or the end of a period of time during which the body has been falling, still is itself a period of time; if it were not, a state of rest, which excludes velocity, would be assumed; velocity can be introduced only according to the space passed through in a period of time and not at the end of a period. — When the differential calculus is actually applied in other spheres of physics where there is no motion at all, for instance in the behaviour of light (apart from its so-called propagation in space) or in the magnitudinal determinations of colours, the first derivative of a quadratic function being here too called velocity, then we must regard this as a still more illegitimate example of the formalism which feigns real existence. —
The motion (says Lagrange) which is represented by the equation s = a P is empirically given in falling bodies ; the next simplest motion after this equation would be that whose equation was s — c P, but nature knows no such motion ; and we do not know what meaning the coefficient c could have. This is true ; but there is a motion whose equation is j* = a P, and this is Kepler’s law of the motion of the bodies of the solar system; — and indeed it would appear an interesting task to show the intended meaning here of the first derivative function
and so on, to treat this equation further and directly by
means of differentiation, and to develop the laws and deter¬ minations of that absolute motion from this starting point, — a task in which analysis might display a brilliance most worthy of itself.
Thus the application of the differential calculus to the elementary equations of motion offers in itself no real interest : formal interest is derived from the general mechanism of the calculus. But another significance is gained by the analysis of motion with respect to the determination of its trajectory : if
this is a curve and its equation contains higher powers, then the transition is necessary of rectilinear functions (as functions of potentiation) to the powers themselves ; the former must be extracted from the original equation of motion, which contains the factor of time, and time must be eliminated ; and so at the same time this factor must be reduced to lower functions of development, from which these equations of linear determina¬ tions can be worked out. This aspect leads us to the interesting element of the other part of the differential calculus.
What has been said so far was said with the purpose of emphasizing and fixing the simple and specific determination of the differential calculus, and of demonstrating it in a few elementary examples. It was seen that this determination consisted in the following process : the coefficient of the member of development (the so-called first derivative) is found from an equation of functions of powers ; this function is a propor¬ tion which is demonstrated in moments of the concrete object; an equation results which determines these moments themselves as between the two proportions. And also we must briefly consider the principle of the integral calculus, and what results from its application for its specific concrete determination. The consideration of this calculus has been simplified and more correctly determined through the fact that it is no longer taken as a method of summation, as it was called in opposition to differentiation ; the increment was there considered the essential ingredient, and with this it appeared in essential connexion with the serial form. — The task of this calculus, as of the differential, is, first, theoretical or rather formal, but, as is well known, it is the converse of the differential. Here a beginning is made from a function which is considered as derivative and as the coefficient of the next member, which originates from the development of an equation as yet un¬ known; and from it the original function of power is to be calculated. The function which in the natural order of develop¬ ment must be regarded as primary is here considered derivative, and that which before was considered derivative here is con¬ sidered as given and in fact as beginning to exist. But it appears that the formal part of this operation has already been per¬ formed by the differential calculus; since in it the transition and relation in general between original function and function
of development is established. In order to apply the function from which we must start, and also in order to effect the transition from it to the original function, it is necessary in many cases to have recourse to the serial form; but it must be remembered that this form as such has nothing to do directly with the peculiar principle of integration.
It appears next that the other part of the task of the calculus, with respect to its formal operation, is the application of the latter. This is now itself the task, namely, to know the meaning (in the sense indicated above), as a separate object, of the original function of the given function, which is considered the first derivative. It might appear that this doctrine was quite done with in the differential calculus; but a further circumstance enters into play which does not allow the matter to be so simple. For it resulted from this calculus that the proportion — which is linear — was obtained from the first derivative of the equation of a curve ; and, therefore, knowing this we also know that the integration of this proportion gives us the equation of the curve in the proportion of abscissa and ordinate; or, if the equation were given for the plane of a curve, then it would be the case that the differential calculus ought already to have taught, with respect to the meaning of the first derivative of such an equation, that this function exhibited the ordinate as function of the abscissa, and, there¬ fore, the equation of a curve.
and if we believe that its ultimately a matter of media contents which shape the worldview of the people, then i think it becomes necessary to post on contemporary social media platforms in order to do propaganda for your causes; we’ve seen how important it is to seize control of media to influence things external to media, and now, we have graciously been given a way to do that. you can and should become a good manipulator
But the question is, which of the determining moments of the object is itself given in the equation ; for the analytic treat¬ ment can proceed only from what is given, and pass thence to the remaining determinations of the object. Thus what is given is not the equation of an area of the curve, nor of the body arising from its revolution, nor of an arc of the curve; but only the proportion of the abscissae and ordinates in the equation of the curve itself is given. Therefore, the transitions from those determinations to this equation itself cannot already be treated in the differential calculus: it is reserved for the integral calculus to find these proportions.
But further it has been shown that the differentiation of an equation of more than one variable magnitude gives the power of development or differential coefficient, not as an equation, but only as a proportion : the task, then, is to indicate in the
moments of the object a second proportion that shall be equal to this first proportion, which is the derivative function. In the integral calculus, on the other hand, the object is the proportion itself of the original function to the derivative (which is here supposed to be given) ; and the task is, to indicate the meaning of the original function (which is to be discovered) in the object of the given first derivative, or rather, since this meaning has already been declared to be the problem (the meaning is, for instance, the plane of a curve, or the curve, imagined as rectilinear, which remains to be rectified; and so on), to demonstrate that such a determination is found by the original function, and to show which is the moment of the object that must be taken for this purpose as initial function of the derivative function.
Now the ordinary method, which uses the idea of the dif¬ ference as equivalent to the infinitely small, makes its task simple : thus, for the quadrature of curves, an infinitely small rectangle, a product of ordinate and element, that is, the infinitely small part of the abscissa, is taken for the trapezium which is supposed to have for one of its sides the infinitesimally small arc opposite to that infinitesimally small part of the abscissa. The product is integrated in this sense, that the integral gives the sum of the infinitely great number of trapezia, the plane whose determination is required, that is, the finite magnitude of this element of the plane. And, similarly, it forms a right-angled triangle out of the infinitesimally small elements of the arc and the ordinates and abscissae belonging to them; in this the square of that arc is supposed to be equal to the sum of the squares of the other two infinitesimally small elements, the integration of which presents us with this arc as a finite arc.
This procedure is based upon the general discovery which is the foundation of this part of Analysis; and here is based upon it in that the quadrated curve, the rectified arc, and so on, stand to a certain function which is given by the equation of the curve, in the relation of so-called original to derivative function. The question now is this: when a certain part of a mathematical object (for instance, of a curve) is assumed to be the derivative function, what other part of it is expressed by the corresponding original function? We know that, when
the function of the ordinate given by the equation of the curve is taken to be the derivative function, then the (relatively) original function expresses the magnitude of the area of the curve which is cut off by this ordinate ; and, when a certain tangential determination is regarded as derivative function, then its original function expresses the magnitude of the arc belonging to this tangential determination, and so on: but the method which uses the infinitesimally small and operates with it does not take the trouble to recognize and to demon¬ strate that these ratios (firstly that which subsists between original and derivative function, and secondly that between the magnitudes of two parts, or conditions, of the mathematical object) together form one proportion. It is the peculiar merit of intellectual insight to have discovered from results already known from without, that certain and specified sides of a mathematical object stand in the relation one to the other of original and derivative function.
In this calculus, of these two the derivative function, or (as it has been determined) the function of potentiation, is the one which is given relatively to the original function ; the latter still remains to be discovered from the former by integration. But it is not given immediately, nor is it given for itself which part or determination of the mathematical object is to be looked at as derivative function, in order to find, through tracing it back to the original function, the other part or determination, whose magnitude the problem requires. The ordinary method, as has been stated, immediately represents certain parts of the object as infinitesimally small, in the form of derivative functions which can be determined from the original equation of the object by differentiation in general ( — thus for the rectification of a curve it takes the infinitesimal abscissae and ordinates) ; and this method takes instead such as can be brought into a connexion with the object of the problem (in the example, with the arc, which also is imagined as infinitesimal) which is fixed and determined by elementary mathematics, by which procedure, these parts being known, that part also of which the magnitude was to be found, is determined. Thus, for rectification, the three infinitesimals we mentioned are connected in the equation of the right-angled triangle, while for quadrature, the ordinate and the infinitesimal
abscissa are connected in a product, a plane being taken, in the general arithmetical manner, as a product of straight lines. The transition from such a so-called element of plane or arc, and so forth, to the magnitude of plane or arc itself now only counts as ascent from the infinitesimal to the finite expression or to the sum of an infinite number of elements, of which the required magnitude is supposed to consist.
We can, therefore, make only the superficial observation that the integral calculus is just the problem of the differential calculus, but inverted, and, in general, more difficult; while the real interest of the integral calculus concerns itself entirely with the relation to each other of the original and the derivative function in concrete objects.
Lagrange nowhere banished the difficulty of any problem in the facile manner of these direct assumptions, nor did he consent to do so in this part of the calculus. It will help the elucidation of the nature of the matter if here too we indicate the detail of his procedure in a few examples. For the task of his method is to demonstrate in itself that a relation of original to derivative function subsists between separate determinations of a mathematical whole, for instance a curve. But in this sphere, because of the nature of the relation itself, this cannot be done in a direct manner, since in the mathematical object the relation connects curved with straight lines, linear dimen¬ sions and their functions with plane dimensions and their function, and so on, — connects, that is, terms which are quali¬ tatively different; the determination thus can only be taken as the mean between a greater and a less. Thus here again there enters spontaneously the form of increment with its plus and minus, and that vigorous “ Developpons ” is here in place; but we have already discussed that purely arithmetical and finite meaning which here is all that belongs to the increments. When we develop the condition that the magnitude which is to be determined must be greater than one and less than another limit (itself easily determinable), we derive such facts as that the function of die ordinate stands in the relation of first derivative function to the function of the area.
my critique here is on the first point, i think that the media is not really simply controlled by some big human on top. i think it’s controlled just as much by the desires of the consumers of media. we’ll get around to 3 eventually, but here i will just say of course the media shapes those desires in turn in a way, but lots of things do
Lagrange’s exposition of the rectification of curves starts from Archimedes’ principle, and is interesting, therefore, as affording an insight into the translation of the Archimedean
method into the principle of modem analysis, which allows us to look upon the inner and true meaning of an affair which by the other mode is pursued but mechanically. The procedure is necessarily analogous to that which has just been indicated: no direct equation results from the principle of Archimedes, which is that the arc of a curve is greater than its chord and less than the sum of two tangents drawn between the ends of the arc and their point of intersection. This fundamental determination of Archimedes is translated into the modem analytic form when an expression is found which is in itself a simple fundamental equation, whereas the form of Archi¬ medes only postulates that there must be an infinite progress between two elements, respectively too great and too small, which each time determine themselves, — this progress ever giving but a new pair of great and small, with an ever narrower limit of inaccuracy. The formalism of the infinitesimally small immediately gives us the equation d t* — d x® + djP. But Lagrange's demonstration, starting from the foundation which has been indicated, shows that the magnitude of the arc stands in the relation of original to a derivative function, in which the characteristic member itself is a function which comes from the relation of a derivative to the original function of the ordinate.
The method of Archimedes, and, at a later date, Kepler’s treatment of stereometrical objects, employ the idea of the infinitesimally small, — a fact which has often been quoted as authorizing the we made of this idea in the differential cal¬ culus, while its peculiarity and distinguishing quality were not sufficiently emphasized. The infinitesimally small means, first, the negation of Quantum as such, that is, of a so-called finite expression, or of that perfect determinateness which belongs to Qpantum as such. And, similarly, the fundamental deter¬ mination in the subsequent famow methods of Valeriw,1 Cavalieri, and others, which are based on the consideration of the relations of geometrical objects, is, that the Qpantum (as such) of determinations which so far are being considered merely as related terms, is to be neglected for this purpose, and consequently they are to be taken as non-magnitudinal.
1 Valerius, Lucas, died 1618 at Rome, called by Galileo die Archimedes of his time : dr quadrature parabolae per simplex faltum.
But here the general affirmative element which is latent in the merely negative determination remains unrecognized and unnoticed — an element which, abstractly, proved to be quali¬ tative magnitudinal determinateness based, more precisely, upon the ratio of powers; — and also, since this relation itself includes a number of more closely determinate relations like powers and the functions of their development, attempts have been made to base these too on the general and negative determination of the same infinitesimally small, and to derive them thence. In Lagrange’s exposition which we have just examined, the determinate affirmative which is contained in Archimedes’ manner of developing the problem has been dis¬ covered, and thus the procedure, which was infected with an unlimited overpassing, has been given its correct limit. The importance of the modern invention in itself, and its capacity to solve problems hitherto intractable, and to treat in a simple manner those which were not insoluble before, is due solely to the discovery of the relation of the original to the so-called derivative, and of the parts which, in a mathematical whole, stand in this relation.
What has been said may serve to make clear what is characteristic in that proportion of magnitudes which is the object of the particular kind of calculus under discussion. Our exposition was able to confine itself to simple problems and the methods of solving them ; and it would neither have been suitable for the determination of the concept (which here alone was our object), nor would it have been in the power of the author, to examine the whole scope of the so-called application of the differential and integral calculus, and to complete the induction that they are based upon the principle which we discovered, by tracing back to it all their problems and their solutions. But our contribution has sufficiently shown that as every other calculus has for object a separate determinateness or relation of magnitude, and addition, multiplication, the raising to powers and extraction of roots, operations with logarithms and with series, and so on, constitute such objects, so too the differential and integral calculus; and the name of relation of a function of powers and of the function of its development or potentiation, is perhaps fittest for whatever is akin to this calculus, for this name places us nearest to an
insight into the nature of the matter. Only, operations accord¬ ing to other magnitudinal relations (addition and so on) are also generally used in this calculus, and so also logarithmic, circular, and serial relations are applied, especially in order to make more tractable expressions for the purpose of the necessary operations of the derivation of original functions from functions of development. The differential and integral calculus has this point of interest in common with the form of the series, that it determines the functions of development which, in series, are called coefficients of the terms; but the interest of this calculus is directed only upon the relation of the original function to the next coefficient of its development, and thus the series tries to represent a sum in the multitude of terms arranged according to powers which have these coefficients. The infinite of the infinite series — that indeter¬ minate expression of the negative element of Quantum in general — has nothing in common with the affirmative deter¬ mination which is contained in the infinite of this calculus. And also the infinitesimally small (whicn appears under the shape of increment), which gives to development the form of series, is a merely external means to this end, and the only meaning of its so-called infinity is to have no meaning except as this methodological device : the series, which in fa't is not what is wanted, produces an excess the removal of which causes the unnecessary trouble. Lagrange’s method too — he took up the serial form again by preference — is hampered by this trouble; although it is in this method that the true peculiarity stands revealed in what is called the application, for, without forcing the forms of d x, dy, and so on, into the objects, that part is directly indicated to which the deter¬ minateness of the derived function (function of development) belongs in them: so that it is clear that here the real matter in hand is not the form of the series.1
* In the above-mentioned criticism ( Jahri . fir wissensch. Krit., II Bd. 1827, Nr. 115, 6 sqq.), interesting views of a sound scholar in this science, Mr. Spehr,* may be found; they are quoted from his Aims Prinzipien da FlutnUnkalkuls, Brunswick, 1826, relating to a fact which he asserts materially to contribute to the obscurities and unscientific parts of the differential calculus, and they agree
* Spehr, Friedrich Wilhelm, 1799-1833, a mathematician of Brunswick: Volls t&nd&gn Lthrbegriff der rtinen Kombinatiorulehre, 1824.
Further Forms connected with the Qualitative Determinateness of Magnitude
what does this change in perspective here do for us? when you feel the media is primarily shaped by the big boy on top, since you’re of course never gonna be a trillionaire, there’s not much you can do but invest your life energy in different men (and sometimes women) who have the capacity to come around and unfuck the media. this is what i like about “YOUR BEING MANIPULATED” in Eddington. deranged right wing maga people, instead of being the least aware of manipulation in media, are genuinely the most aware of it. this is a core feature; i think awareness of manipulation in the context of a manipulation / propaganda model of media, contributes to one’s microfascist tendencies, the desire for the fascist leader etc…
In the differential calculus the infinitesimally small appears in its affirmative meaning as qualitative magnitudinal deter¬ minateness; and of this it has further been shown that it is present in this calculus not only as determinateness of powers in general, but more especially as the determinateness of the ratio of a function of powers to the power of development. The qualitative determinateness is also present in a wider and, so to say, weaker form, which, together with the connected employment of the infinitesimally small and its meaning in this employment, will be considered in this Observation.
We begin from what precedes, and must first remark in this regard that the various determinations of powers, in the analytic aspect, manifest themselves as purely formal and quite homogeneous, since they denote numerical magnitudes, which, as such, do not possess this mutual relation of qualitative difference. But when applied to objects of space, the analytic relation shows itself in its qualitative determinateness as the transition from linear to plane determinations, from deter¬ minations of straight lines to determinations of curves, and so
with what has been said about the general nature of the theory of this calculus. “Purely arithmetical investigations,” he says, “(admittedly related more closely than any others to the differential calculus) were not distinguished from the differential calculus proper, and have ever been confused (as by Lagrange) with the matter itself, which latter was regarded as a mere 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 reverse is the case, and these applications are the object of the real differential calculus, which, from the analysis, presupposes all these arithmetical develop¬ ments and operations.” — We have shown how with Lagrange the separation of the so-called application from the procedure of the general part, which starts from series, serves only to emphasize the peculiar nature of the differential calculus for itself. And it is strange that the author, having this interesting under¬ standing that the so-called applications are just what constitutes the object of the differential calculus proper, should enter upon the formal metaphysics (there quoted) of Continuous Magnitude, Becoming, Flow, and so on, and should even desire to add new ballast to this old. These determinations ar c formal because they are only general categories, which do not indicate what is specific in the matter which was to be learned and abstracted from the concrete doctrines, the applications.
on. This application further involves that spatial objects which from their very nature are given in the form of continuous magnitudes, are taken as discrete, the plane as a multitude of lines, the line as a multitude of points, and so on. This solution is interesting only in one respect, which is that it itself determines the points into which the line is analysed and the lines into which the plane, and so forth, in order that from such a determination it may proceed analytically, which really means arithmetically; the starting-points for the mag- nitudinal determinations which are to be found are those elements whence are to be derived the function and equation for the concrete, that is, the continuous magnitude. In those problems which are interesting chiefly because they employ this procedure, something which is determinate in itself is de¬ manded, in the element, for the starting-point, — in opposition to the method which is indirect, because it can begin on the contrary only with limits between which is supposed to lie that entity, determinate for itself, which is its objective. In both methods the result is the same if only it is possible to discover the law of further determination and impossible to reach the perfect (that is, so-called finite) determination which is demanded. To Kepler is ascribed the honour of first having thought of this reversal of the progress, and of making the discrete its starting-point. He expresses this simply when he explains how he understands the first theorem in Archimedes’ cyclometry. Archimedes’ first theorem is, of course, that a circle is equal to a right-angled triangle having one catheter equal to the radius and one equal to the circumference of the circle. Kepler takes the meaning of the theorem to be this, that the circumference of the circle has as many parts as it has points, that is, infinitely many,- each of which may be considered as the base of an isosceles triangle, and so on: he thus gives expression to the dissolution of the continuous into the form of discreteness. The expression “infinite” which here occurs is still far distant from the determination which it is destined to have in the differential calculus. — A determinateness or func¬ tion having now been found for such discreta, they must next be united, and exist essentially as elements of the continuous. But a sum of points produces no line and a sum of lines no olane: the points are, therefore, immediately taken as linear
and the lines as plane in nature. But also these linear entities must not ye; be lines (which they would be if they were taken as quantum) : they are, therefore, imagined as infinitesimally small. But what is discrete can be united only externally, the moments retaining their meaning of discrete Ones; the analytic transition from them is made only to their sum, and is not also the geometric transition from point into line, line into plane, and so on; and, therefore, the element which has its determination as point or as line is given the quality of line with the former, and the quality of plane with the latter determination, so that the sum, being a sum of little lines, may become a line, and being a sum of little planes, a plane.
The need of acquiring this moment of qualitative transition and of recurring to the infinitesimally small to this end must be regarded as the source of all the ideas which, intended to overcome this difficulty, are the greatest difficulty themselves. In order that one might, dispense with this expedient, it would have to be possible to show that the analytic procedure itself, which appears as a mere summation, in fact already contains multiplication. But in this regard a new assumption enters, which constitutes the basis of this application of arithmetical relations to geometrical figurations. This assumption is,, that arithmetical multiplication is a transition to a higher dimension for the geometrical determination too, and that the arith¬ metical multiplication of magnitudes which, according to their spatial determinations, are lines, also extracts a plane from the linear determination. Three times four linear feet are twelve linear feet, but three linear feet times four linear feet are twelve plane feet (square feet), for the unit in both, since each is a discrete magnitude, is the same. The multiplication of lines by lines at first appears meaningless, since multiplication only deals with numbers; that is, it is a change of entities which are perfectly homogeneous with that into which they pass over (the product), and change only their magnitude. On the other hand, a process of multiplying a line as such by a line — which has been called ductus liruae in lineam , like plant in planum, and is also ductus puncti in lineam — is a change not only of the mag¬ nitude but also of the line as qualitative determination of spatiality, as a dimension; the transition of line into plane must be taken as its sclf-extemalization , which for the point
is a line, and for the plane, a volume. It is this process which is imagined when it is said that the movement of a point is a line, and so forth; but movement includes the determination of time, and in this idea, therefore, appears rather as a con¬ tingent and external variation of the condition ; whereas it is the conceptual determinateness (which was expressed as self- extemalization) which must be taken, — that qualitative change, which in arithmetic is the multiplication, of Unit (the point and so on) into Amount (the line and so on). — We may here make this further remark, that in the self-externalization of the plane (which would manifest itself as a multiplication of plane into plane) the appearance of a difference between arithmetical and geometrical products results when the self- externalization of the plane, as ductus plani in planum, would in arithmetic produce the multiplication of two determinations of the second degree, that is, a product of four dimensions, which in the geometrical determination is however reduced to three. Although on the one hand Number, because it has One for principle, provides the fixed determination for the external quantity, yet equally its productive power is formal. Taken as a numerical determination, 3x3, when it repro¬ duces itself, is 3 x 3 X3 x 3; but this same magnitude as plane determination is not allowed, when it reproduces itself, to proceed beyond 3x3x3, because space, imagined as a progress from the point, or the merely abstract limit, has its true limit as concrete determinateness after line in the third dimension. This difference might prove powerful with regard to free movement, wherein one (spatial) side is governed by geometrical determination (in Kepler’s law j3 : fi), and the other (temporal) side by arithmetical determination.
It will now be evident without further remark how the qualitative here considered differs from the subject of the previous Observation. There the qualitative element consisted in the determinateness of powers : here, like the infinitesimally small, it stands in the mere arithmetical relation of factor to product, or as point to line, line to plane, and so on. And the qualitative transition which must be made to the continuous from the discrete (into which continuous magnitude is imagined as dissolved) is effected by a process of summation.
a multiplication and, therefore, a transition from linear to plane determination; and this appears most simply in the manner in which (for instance) it is shown that the area of a trapezium is equal to the product of the sum of the two opposite parallel lines and half the height. This height is imagined simply as the amount of a multitude of discrete mag¬ nitudes which must be summed up. These magnitudes are lines which lie parallel between those two limiting parallels; their number is infinite, for they must constitute the surface, and also they are lines which, therefore, in order to be of plane nature, must be posited together with negation. In order to escape the difficulty that a sum of lines is to produce a plane, lines are immediately assumed to be planes, but infinitely narrow, for their only determination lies in the linear element of the parallel limits of the trapezium, being parallel, and being limited by the other pair of rectilinear sides of the trapezium, they can be imagined as terms of an arithmetical progression, having a uniform difference which, however, need not be determined, and having those two parallels for first and last terms ; the sum of this series is of course the product of the parallels and half the amount of the terms. This latter quantum is called Amount only relatively to the idea of the in finitely many lines: it is the magnitudinal deter¬ minateness of something continuous, namely height. Clearly that which is called sum is also a ductus lineae in lineam, multi¬ plication of linear by linear element, and, therefore, the produc¬ tion (according to the above determination) of something of plane nature. In the simplest case of any rectangle A B, each of the two factors is a simple magnitude; already in the next example (itself elementary) of the trapezium, only one factor is once half the height, while the other is determined by a progression : it too is of linear nature, but the determinateness of its magnitude is more complex : and since it can be expressed only by a series, the endeavour to sum it up is called analytical, that is, arithmetical ; while multiplication here is the geomet¬ rical moment, the qualitative part of the transition from the dimension of line into plane : one factor is taken, discretely, only for the arithmetical determination of the other, and, like the other, is itself the magnitude of a linear something.
The method in which planes are imagined as sums of lines
i think that when we understand that our desire has some little power in this way, the affordance is not to plug our desire into the big leader who will fix the media for us, but instead to plug it back into our minority communities
is also frequently used when multiplication as such is not used in order to find the result. This happens where the object is not to indicate the magnitude in the equation as Quantum, but as a proportion. For instance, it is commonly proved that the area of a circle is to the area of an ellipse, of which the major axis is equal to the diameter of the circle, as the major axis is to the minor axis, by assuming that each of these areas is the stun of the relative ordinates. Each ordinate of the ellipse is to the corresponding ordinate of the circle as the minor is to the major axis : it is concluded that, therefore, the sums of the ordinates too (that is, the areas) are in the same proportion. Those who here wish to avoid the idea that an area is a sum of lines have recourse to the common and quite gratuitous makeshift of making the ordinates into trapezia of infinitely small breadth: the equation is only a proportion, and there¬ fore only one of the two linear elements of a plane enters into the comparison. The other element (the axis of the abscissa) is assumed to be equal in ellipse and in circle, and therefore as = x in so far as it is a factor of arithmetical magnitudinal determination: the proportion therefore depends solely upon the relation of the one determining moment. The two dimen¬ sions are essential to the idea of area: but the magnitudinal determination, as it is required to be indicated in this pro¬ portion, concerns itself only with one moment ; and when the idea of sum is added to this one moment (which is a surrender to, or attempted propping of, the idea), then this means that the real point demanded by mathematical determinateness is here missed.
This exposition also contains the criterion of Cavalieri’s method of the indivisible (which has been mentioned above) : this, too, is therefore justified and need not have recourse to the infinitesimally small. These indivisibilia are lines when he is considering a plane, and squares or plane circles when he is considering a pyramid, cone, and so on. The basic line or plane (which he takes as determinate) is called the regula ; it is the constant, and with reference to a series it is its first or last term; and to it these indivisibilia are considered parallel, that is, as having the same determination with regard to the figure. Now Cavalieri’s general fundamental proposition is ( Exerc . Geometr. VI., in the later work Exerc. /., p. 6), “that all
figures, both plane and solid, are proportionate to all their indivisibilia, and compare these collectively or, when there is a common proportion in them, distributively.”— For this pur¬ pose he compares, in figures of the same base and height, the proportions between lines drawn parallel to the former and at equal distance from it; all such lines in a figure have one and the same determination and constitute its whole content. In this manner, too, Cavalieri proves, for instance, the elemen¬ tary proposition that parallelograms of equal height are pro¬ portionate each to its base : any two lines drawn at an equal distance from the base and parallel to it in the two figures are in the same proportion to the bases; and so, therefore, are the whole figures. In fact, the lines do not constitute the content of the figure in so far as it is continuous , but only in so far as it is to be determined arithmetically; the linear factor is its element, and through it alone its determinateness must be seized.
And here we are led to reflect upon the difference which exists with respect to that element in which the determinateness of a figure consists; it is either of the same nature as is here the height of the figure, or it is external limit. In so far as it is determinateness as being external limit, it is admitted that the continuity of the figure follows, so to speak, upon the equality or the proportion of the limit; thus the equality of figures which coincide with one another follows from the feet that the limiting lines coincide with one another. But when parallelograms are of equal height and base, only the latter determinateness is external limit; height, and not parallelism (and on this the second capital determination of figures, that is their ratio, depends), introduces a second principle which determines external limits. Euclid’s proof that parallelograms of equal height and base are equal, reduces them to triangles, that is, to continua limited externally: in Cavalieri’s proof, and first in his proof of the proportionality of parallelograms, the limit is magnitudinal determinateness as such in general, which is explained to be taken as applied to each pair of lines drawn at an equal distance in the two figures. These lines, which are either equal or in an equal ratio with the base, being „ taken collectively, produce the figures which also are in the same ratio. The idea of ai> aggregate of lines is incompatible
with the continuity of a figure; the mere consideration of lines quite exhausts the essential deterxninateness. Cavalieri frequently answers the objection that the idea of the indivisible necessarily involves the comparison of lines or planes infinite in amount ( Geom . Lib. II. Prop. I. Schol.) ; and makes the just distinction, that it is not their amount, which we do not know ( — it is, as has been observed, an empty auxiliary idea — ), but only magnitude, that is quantitative determinateness as such, equal to the space enclosed by these lines, which he compares. This space is contained in limits, and, therefore, its magnitude too is contained within these limits : the continuous is nothing other than the indivisibilia themselves , says Cavalieri : if it were outside them it would be incommensurable: but it would be absurd to say that limited continua were incommensurable.
Clearly, thus, Cavalieri attempts to distinguish what belongs to the external existence of the continuous from that which constitutes its determinateness and needs to be emphasized only for comparison and for theorems dealing with it. It is true that the categories which he uses for this purpose (the fact that the continuous is composed or consists of the indivisibilia, and the like) are inadequate, since they also involve the intuition, or, as was said before, the external existence, of the continuous ; instead of saying that “the continuous is nothing but the indivisibilia themselves,” it would be more correct and therefore also immediately more clear to say that the mag- nitudinal determinateness of the continuous is the same as that of the indivisibilia themselves. — Cavalieri does not care for the faulty conclusion that there are greater and less infinites, which (he says) is drawn by the schools from the idea that the indivisibilia constitute the continuous ; and, further, (Geom. Lib. VII. Preej.) he expresses the more definite knowledge that his method of proof by no means forces him into the idea of the composition of the continuous from indivisibilia : the con¬ tinua only follow the proportion of the indivisibilia. He claims to have taken the aggregates of indivisibilia, not in that manner in which, for the sake of an infinite multitude of lines or planes, they appear to fall into the determination of infinity, but in so far as they contain a determinate condition and nature of limitedness. But in order finally to remove this 1 tumbling-block, he does not spare himself the pains of proving
the capital propositions of his geometry (in the seventh book, specially added for this purpose) in a manner designed to be uninfected with infinity. — This manner reduces the proof to the ordinary form (mentioned before) of the coincidence of figures, that is, as was observed, to the idea of determinateness as external spatial limit.
We may further remark about this form of coinciding, that altogether it is what may be called a childish aid for sensuous intuition. In the elementary theorems about triangles, two triangles are imagined side by side; of their six component parts, three are assumed equal to the corresponding three in the other triangle, and then it is demonstrated that such triangles are equal in all respects; that is, that each has the remaining three parts also equal to those of the other, because it follows from the equality of the first three that they coincide. If the matter be taken more abstractly, it is just because of this equality of each pair of the parts corresponding to each other in both, that there is only one triangle, in which three parts are taken as already determinate, whence follows the determinateness of the three remaining parts. In this manner determinateness is demonstrated as complete in three parts; and, for determinateness as such, the three remaining parts represent a superfluity, the superfluity of sensuous existence, that is, of the intuition of continuity. Expressed in this form, qualitative determinateness manifests itself as distinct from the object of intuition, which is the whole as continuous in itself; and coincidence does not allow this distinction to be perceived.
With parallel lines and parallelograms there comes into play (as has been observed) a new circumstance, namely, partly the equality of the angles only and partly the height of the figures, from which latter their external limits (the sides of the parallelograms) are distinct. Here a doubtful question arises, that is, how far in these figures— apart from the deter¬ minateness of one side, the base, which is an external limit— we are to take for the other determinateness (a) the other external limit, namely, the other side of the parallelogram, or (b) the height. Where there are two such figures having the same base and height, the one being rectangular and the other having very acute angles (the opposite pair being there¬ fore very obtuse), the latter could easily appear greater to
intuition than the former, in so far as it might take its long side as determinant and (according to Cavalieri’s method of imagining the matter) might compare the areas according to the multitude of parallel lines which can pass through them : the longer side might be considered as affording a potentiality of more lines than the vertical side of the rectangle affords. Such an idea, however, provides no objection to Cavalieri’s method ; for the multitude of parallel lines imagined for pur¬ poses of comparison in the two parallelograms presupposes their equidistance from one another or from the base: it follows, therefore, that height and not the other side of the parallelogram is the other determining moment. But this changes further when two parallelograms are compared with each other which have the same height and base but are not in the same plane, and are at different angles with a third plane: here the parallel sections which arise when the third plane is imagined as passing, through them and moving parallel to itself, are no longer equidistant from one another, and the two planes are unequal. Cavalieri is very careful to draw attention to this distinction, which he defines as the distinction between a transitus rectus and a transitus obliquus of the indi- visibilia (already in Exercit. I. n. XII. sqq. and also in Geometr. I. II.), and prevents a superficial misunderstanding which might arise in this direction. Barrow in his above-cited work (Led. Geom. II. p. 21) also uses the method of indivisibilia, although he adulterates it with the assumption (which from him passed on to his pupil, Newton, and to other mathematical contem¬ poraries, among them Leibniz) that a curvilinear triangle, like the so-called characteristic triangle, may be equated with a rectilinear triangle in so far as both are infinitely, that is, very small; and I remember that he quotes an objection of Tacquet,1 an ingenious geometer who worked upon what then were new methods, which points to that same end. The difficulty which he raises refers to the question, which line, in the calculation of conical and spherical surfaces, should be taken as fundamental moment of determination for the con¬ sideration based upon application of the discrete. He says that Tacquet’s objection to the method of indivisibilia is this, that
now for critique of the second point: the media is used instrumentally for goals external to the media. here, i think that the media can mostly be used for the internal functioning of the media. it has some marginal effects on things immediately next to it, but mostly it just propagates itself
1 Tacquet, Andr., 1611-1660, Professor in the Jesuit College at Antwerp: Cylindriconm it amudarium libri V, 1651-9.
in the calculation of the surface of a right-angled cone the triangle, of the cone is imagined, in this atomistic method, as composed of straight lines parallel to the base and at right angles to the axis, which also are the radii of the circles of which the surface of the cone consists. And if this surface is determined from the sum of the circumferences, and this sum from the number of their radii, that is, from the magnitude of the axis, or the height of the cone, then (he proceeds) such a result is in contradiction with the truth elsewhere taught and demonstrated by Archimedes. And Barrow shows that, to determine the surface, it is not the axis but the side of the triangle of the cone which must be taken as the line the revolution of which produces the surface ; this line, therefore, and not the axis, must be taken as the magnitudinal deter* minateness for the multitude of circumferences.
Such objections and hesitancies have their sole origin in the indeterminate idea which they use of the infinite multitude of points of which a line, or of lines of which a plane, and so forth, is regarded as consisting: the essential magnitudinal determinateness of lines and planes is obscured by this idea. — It has been the intention of these Observations to make evident the affirmative determinations which so to speak remain in the background in the varied use which is made of the in¬ finitesimally small in mathematics; and to raise them from the obscurity in which they are wrapped by this purely negative category. In the infinite series (as in Archimedes’ cyclometry) the Infinite means nothing more than that the law of further determination is known, but that the so-called finite (that is, arithmetical) expression, is not given, so that it is impossible to effect the reduction of the curve to the straight line : this incommensurability constitutes their qualitative difference. The qualitative difference between discrete and continuous in general also contains a negative determination, which causes them to appear incommensurable and introduces the infinite in this sense, that the continuous (which is to be taken as discrete) must now have no Quantum according to its con¬ tinuous determinateness. The continuous (which, arithmeti¬ cally, is to be taken as product) is thus posited discretely in itself, as being analysed into the elements which are its factors : in these lies its magnitudinal determinateness ; and just because
they are these elements or factors, they are of a lower dimension, and, in so far as there is any determinateness of power, of a lower power than the magnitude of which they are elements or factors. Arithmetically this difference appears as merely quantitative, as a difference of root or power or any other determinateness of power; but when the expression concerns only the quantitative as such (for instance a : a2 or d. a 2 — 2d : a2= 2 : a, or, for the law of gravitation, t : at2), the ratios which result are meaningless (i : a, i : a, i : a t) ; the sides ought to be held apart against their merely quantitative determination by a difference in qualitative meaning, as siat2, where the magnitude as a quality is expressed as function of the magnitude of another quality. Here then it is only quantitative determinateness which is present to con¬ sciousness ; with it, it is easy to operate according to its manner, and no objection can be offered if the magnitude of one line be multiplied by the magnitude of another; but from the multiplication of these same magnitudes there also results the qualitative change of the transition of line into plane : and in so far a negative determination comes into play. It is this that causes the difficulty which is solved by an understanding of its peculiarity and of the simple nature of the matter, but by the help of the infinite, which is intended to remove it, is suspended quite unsolved and in confusion.
The Infinity of Quantum has proved itself to be the negative Beyond of Quantum; but it belongs to Quantum. This Beyond is the Qualitative in general. Infinite Quantum, as the union of the two moments (quantitative and qualitative determinate¬ ness), is Ratio.
In Ratio the determinateness of Quantum no longer is indifferent: it is qualitatively determined as absolutely related to its Beyond. It continues itself into its Beyond ; and this, for the present, is simply any other Quantum. But essentially they are related to one another not as external Quanta; each has its determinateness in this relation to the other. In this their other¬ ness they have thus returned upon themselves ; what each is, it is in the other; and the other constitutes the determinateness of each. — Thus the passing of Quantum beyond itself now has this meaning, that it did not merely change into an Other, nor into its abstract Other (its negative Beyond), but that it there reached its determinateness : it finds itself in its Beyond, which is another Quantum. The quality, or conceptual determinate- ness, of Quantum is its externality in general. Now in Ratio it is posited as having its determinateness in its externality and in another Quantum, and as finding its true nature in its Beyond.
The entities that have the relation which we have found, are Quanta. This relation is itself a magnitude; Quantum is posited, not only as being in a Ratio, but as being itself a Ratio; it is a Quantum in general which contains this quali¬ tative determinateness within itself. Taken thus as Ratio, it expresses its existence as self-contained totality and its indiffer¬ ence to limit, through the fact that it has within itself the externality of its determinateness and in this externality is related only to itself, that is, is in itself infinite.
(i) Direct Ratio. Here the qualitative element does not show forth for itself as such: it is in a purely quantitative
manner that Quantum is posited as having its determinateness in its externality. — In itself Quantitative Ratio is the con¬ tradiction of externality and of self-relation, of the existence of Quanta and of their negation; — it transcends itself when next
(a) in Indirect Ratio there is posited the negation as such of one Quantum together with the alteration of the other, and the alterability of Direct Ratio itself;
the view of the media as this all-powerful meta-institution originates from a belief that ideas are the drivers of history. i don’t disagree that they play a part, but it’s just a bit absurd if you believe in ideas as primary if trump’s approval rating is so low that they stopped doing polls, and yet he’s totally able to continue functioning without any issues. there’s a lot more that goes into revolutions than there simply being a widespread idea of a better way. how often do you program something and you have an idea of a better way of doing things, but the material cost of revolutionizing your code is too high, so it never happens. ideas are cheap and the very first step
(3) in Ratio of Powers, however, unity, which in its differ¬ ence is self-related, asserts itself as simple self-production of Quantum ; this qualitative element being finally posited in simple determination and as identical with Quantum, becomes Measure.
— In the preceding Observations, which deal with the infinite of Quantity (that is, with the qualitative moment in Quantity), much that is true of the nature of the following Ratios has been anticipated. All that remains, therefore, is to explain the abstract concept of these Ratios.
1. In that Ratio which is immediate and, therefore, direct, the determinateness of one Quantum is contained reciprocally in the determinateness of the other. There is only one deter¬ minateness or limit of the two, which is also itself a Quantum, and this is the exponent of the Ratio.
2. The exponent may be any Quantum; but it is a quali¬ tatively determinate Quantum self-related in its externality only in so far as it has in itself its difference, its Beyond and otherness. But this difference of the Quantum in itself is the difference of Unit and Amount : Unit is self-determinateness ; Amount, the indifferent oscillation about determinateness, the external indifference of Quantum. At first Unit and Amount were moments of Quantum; now, in Ratio (which to this extent is Quantum realized) each of its moments appears as an independent Quantum, and as determinations of its existence, as limitations of its otherwise purely external and indifferent magnitudinal determinateness.
The exponent is this difference as simple determinateness, that is, it immediately contains in itself the meaning of both determinations. It is, first, Quantum, and thus is Amount; if one side of the Ratio, which is taken as Unit, is expressed as numerical One — and as such only it is counted, — then the other, Amount, is the Quantum of the exponent itself. Secondly, it is simple determinateness as the qualitative element of the sides of the Ratio : if the Quantum of one is determined, the Quantum of the other too is determined by the exponent, and it is quite indifferent how the first is determined : as Quantum determinate for itself it no longer has a meaning, but can equally well be any other without changing the determinate¬ ness of the Ratio, which rests wholly upon the exponent. The one, which is taken as Unit, always remains Unit, however great it becomes; and the other, however great it likewise becomes in the process, must always remain the same Amount of that Unit.
3. Thus really both only constitute one Quantum; one relatively to the other has the value only of Unit and not of an Amount, and the other has only that of an Amount; and thus, according to their conceptual determinateness, they are not themselves complete Quanta. But this incompleteness is a negation in them, — not a negation which follows from their general alterability according to which one (and each is one of the two) can assume any magnitude, but according to the determination that when one is altered the other increases or decreases proportionally; this means (as has been shown) that only one (Unit) changes as Quantum ; the other side (Amount) remains the same Quantum of units, and even the former has meaning only as Unit, however much it may change as Quan¬ tum. Thus each of the two sides is only one of the two moments of Quantum, and that independence which is its peculiarity, is negated in itself: in this qualitative connexion they must be posited as negative to each other.
The exponent is supposed to be the complete Quantum, for in it the determination of the two sides coincides ; but in fact, as quotient, it only has the value of the Amount, or of the Unit. There is here no determination showing which of the sides of the ratio must be taken as Unit and which as Amount; if one (the Quantum B) be measured against the Quantum A
as Unit, then the quotient C is the Amount of such Units ; but if A itself be taken as Amount, then the quotient C is the Unit which the Amount A demands for the Quantum B; thus this quotient, as exponent, is not posited as what it is supposed to be, namely, the determinant of the ratio or its qualitative unity. It is posited as such only in so far as it has the meaning of being the union of both moments, of Unit and Amount. These sides are present as Quanta (such as they are supposed to be in explicit Quantum, which is Ratio), but also only in the value which they are supposed to have as being sides of the Ratio, — which means that they are incomplete Quanta and count only as one of these qualitative moments; and, therefore, they must be posited as having this their negation. Thus there arises a Ratio more real and better corresponding to its determination; the exponent here has the meaning of being their product; Ratio according to this determinateness is Inverse Ratio.
I. The Ratio which has now been reached is transcended Direct Ratio; the first stage was immediate and, therefore, not truly determinate Ratio; here determinateness has been added, in such a manner that the exponent counts as product, as the unity of Unit and Amount. With regard to immediacy, it could indifferently be taken as Unit and as Amount, as was shown above, and so it also existed as Quantum only in general, and hence, by preference, as Amount ; one side had to be taken as Unit and as One, to which the other stood in the relation of fixed Amount, which also was the exponent; its quality, therefore, was only this, that this Quantum is taken as fixed, or rather that what is fixed has the meaning only of Quantum.
Now in Inverse Ratio, too, the exponent, as Quantum, is taken as immediate; and any Quantum can be taken as fixed. But this Quantum is not related as fixed Amount to the One of the other Quantum in the ratio ; this ratio, which before was fixed, is now posited as variable; if another Quantum is taken as the One of the one side, then the other no longer remains the same amount of units of the first. In Direct Ratio
i got sidetracked, forgive me… so if media largely produces more media, how does this affect our course of action? i think it means it is necessary to understand that 1, growth of media power in itself is not worthwhile, (unless you’re simply interested in media,) so 2, we must keep in mind how what we’re actually doing compares to what we’re trying to achieve as we’re growing media power; we can’t just defer it to the end. so in practice, with brain.worm.sh for instance, if the goal was political action, it would not be correct to simply worry about growing it right now, so that later we can be like elon musk and deviously exploit it for our goals. that doesn’t work if material infrastructure isn’t built along the way outside of the media platform. this material infrastructure off to the sides is what actually grants us capacity to achieve the external goal. i hope it makes sense
this unit is only the common part of both sides; as such it con¬ tinues itself into the other side, which is Amount; and Amount itself for itself, or the exponent, is indifferent to the Unit.
But under the present determinateness of Ratio, Amount as such is varied relatively to the One to which it is related as the other side of the Ratio. It becomes another when another One is taken as Quantum. Consequently, though the exponent is only immediate and a Quantum arbitrarily taken as fixed, yet it does not preserve itself as such in the side of the Ratio : it, and with it the direct ratio of the sides, is variable. Thus in the present Ratio the exponent as determining Quantum is posited as negating itself as Quantum of the Ratio: hence it is posited as qualitative, as Limit, so that the qualitative element emphasizes itself in opposition to the quantitative. — In Direct Ratio the variation of both sides is the single altera¬ tion only of the Quantum which the unit (the common element) is supposed to be; one side is increased or decreased in the same proportion as is the other; the Ratio itself is indifferent to this variation, which remains external. But in Inverse Ratio the variation, although likewise arbitrary according to the indifferent quantitative moment, is contained within the Ratio; and even this arbitrary quantitative overpassing is circum¬ scribed by a limit, namely, the negative determinateness of the exponent.
2. This qualitative nature of Indirect Ratio must be further considered — in its realization; and the complication of the affirmative with the negative which it contains must be analysed. — Quantum is posited as qualitatively determining Quantum, that is, itself, as representing itself in itself as its own limit. Thus, first, it is an immediate magnitude as simple determinateness; it is the whole as existent and affirmative Quantum. Secondly, however, this immediate determinateness is also limit; and hence it is differentiated into two Qpanta, which stand primarily in the relation of Other to each other ; — but, since it is their qualitative and their complete deter¬ minateness, it is the unity of Unit and Amount ; it is product, and they are its factors. Thus partly the exponent of their ratio is identical with itself in them and is their affirmative, and thus they are Quanta; and partly, as negation posited in, them, it is the element of unity in them, according to which
each is an immediate and limited Quantum, and limited in such a way that it is identical with its Other only in itself. Thirdly, as simple determinateness, it is the negative unity of this its differentiation into two Quanta and the limit of their reciprocal limitation.
According to these determinations, the two moments limit each other in the exponent; and since the exponent is their determinate unity, each is the negative of the other; one increases in the same ratio in which the other decreases ; each has its magnitude in so far as it is in contact with the magnitude of the other and is what the other lacks. Thus each negatively continues itself into the other; it cancels its own Amount in the other, and is what it is only by virtue of the negation or limit which the other posits in it. Thus each contains and has its measure in the other, for each is to be only that Quantum which the other is not; the magnitude of the other is indis¬ pensable to the value of each, and therefore inseparable from it.
This continuity of one in the other constitutes the moment of unity by which they form a Ratio, — or the one determinate¬ ness and simple limit which is the exponent. This unity, the whole, constitutes the Being-in-Self of each, which is distinct from their magnitude as it is found: according to the latter, each exists only in so far as it withdraws from the other part of their common Being-in-Self or whole. But it can withdraw from the other only as much as makes it equal to this Being- in-Self; it has its maximum in the exponent, which, according to the second determination which we indicated, is the limit of their reciprocal limitation. And since each is moment of the Ratio only in so far as it limits the other and therefore is limited by it, it loses this its determination by equating itself with its Being-in-Self; in it not only the other magnitude becomes nought, but it vanishes itself, since it is not mere Quantum, but is to be what it is as such only as being such a moment of the Ratio. Thus each side is the contradiction of the determination as their Being-in-Self, that is, of the unity of the whole (which is the exponent), and of the determination as moment of the Ratio : and this contradiction is once more infinity in a new and peculiar form.
The exponent is limit of the sides of its Ratio, within which they increase and decrease relatively to each other ; and accord-
ing to the affirmative determinateness which the exponent (as Quantum) is, neither can become equal to it. As limit thus of their reciprocal limitation, it is (a) their Beyond, to which they can approximate infinitely; but they cannot reach it. This infinity in which they approximate to it is the bad infinity of the infinite progress; this itself is finite, and has its barrier in its opposite, in the finitude of the two sides and of the exponent; it, therefore, is only approximation. But (fi) bad infinity is here also posited as what in truth it is, namely as the merely negative moment in general, according to which the exponent is simple limit as Being-in-Self in relation to the distinct Qjianta of the Ratio ; their finitude is referred to this as to a pure variable, although it remains quite distinct from them as their negation. This infinite, to which they can but approximate, further is found and is present as affirmative Hither : this is the simple Quantum of the exponent. Here the Beyond is reached with which the sides of the Ratio are tainted : it is in itself the unity of both and, therefore, in itself the other side of each; for each only counts for as much as the other does not, and thus the whole determinateness of each resides in the other; and this their Being-in-Self is, as affirmative infinity, simply the exponent.
3. But now the result is that Inverse Ratio has passed over into a determination different from that which it had at first. The determination was, that a Quantum (as immediate) is related also to another Quantum in such a manner that it is greater in the proportion in which the other is smaller, that it is what it is by virtue of its negative attitude to the other; and similarly a third magnitude is the common barrier of this their increase. Here this variation (as opposed to the Qualita¬ tive as fixed limit) is their peculiarity: they have the deter¬ mination of variable magnitudes, having this fixed limit as an infinite Beyond.
We have found and must coordinate the determinations that this infinite Beyond exists as a present and finite (but optional) Quantum; and, further, that its fixity (by virtue of which it is thus related to the quantitative as infinite Beyond) which is the qualitative element of Being only as abstract self-relation, has developed itself with itself as mediation of itself in its Other, namely the finite moments of the Ratio. The universal
to elaborate a bit more, suppose i am the good manipulator with a platform of 1000 users, and now i want to stage some irl protest or whatever. the way how things actually get done is along the way to creating a platform of this size adjacent small reading groups, signal chats, affinity groups, whatever form and link up, especially geographically centered, and they have their own irl meetups. and all of this is taking place mostly outside of the platform. maybe the platform facilitated initial contact. if stuff like this occurs, then as long as manipulation occurs which is aligned with the capacities of these structures, then that’s how things really get done. if, on the other hand, this is the first time anything external to the medium is taking place, and i’m just trying to manipulate people through this one to many channel of the media, and it’s totally stratified, grapified, there are no external relationships, there’s simply no way for anything to occur outside the platform. by the time the material infrastructure will have been developed, the moment for revolution has already passed
part of this is contained in the fact that the whole as exponent is the limit of the reciprocal limitation of the two members; thus negation of negation and, therefore, infinity (an affirma¬ tive attitude to itself) is posited. A closer determination is, that in itself the exponent is product and, therefore, unity of Unit and Amount, whereas each of the two members is only one of these moments ; it, therefore, includes them, and in them (as containing them) relates itself to itself. But in Inverse Ratio the difference develops into the externality of quantitative Being; and the Qualitative is not merely what is fixed or immediately includes the moments, but is found coalescing with itself in self-external otherness. It is this determination which manifests itself as result in the moments which we discovered. For the exponent is found to be Being-in-Self, the moments of which are realized in Quanta and in their vari¬ ability in general; the indifference of their magnitudes in their variation manifests itself as infinite progress, and this is based upon the fact that in their indifference it is their deter¬ minateness that they have their value in the value of an other. Thus (a) in the affirmative aspect of their Quantum they are, in themselves, the whole of the exponent. Equally, (0) their negative moment, for their reciprocal limitation, is the mag¬ nitude of the exponent; their limit is that of the exponent. The infinite progress of their determinate existence and of their limitation, and the negation of any particular value, implies that they have no other immanent limit or fixed immediacy. This negation, therefore, is the negation of the self-externality of the exponent which is represented in them ; and the exponent (taken as Quantum in general and as analysed into Quanta) is posited as that which preserves itself amid the negation of their indifferent persistence and collapses into itself, and, there¬ fore, as the determinant of any such transition beyond itself.
Hence Ratio has come to be determined as the Ratio of Powers.
i. Quantum, which in its other-being identifies itself with itself and determines its self-overpassing, has reached Being- for-Self. It is thus qualitative totality, and, positing itself as
developed, it has for moments the conceptual determinations of number — Unit and Amount. Even in Inverse Ratio the latter is a multitude determined, not by the former as such, but from without by a third element: now it is posited as determined only by the former. This happens in the Ratio of Powers, where Unit, which in itselfis Amount, is also Amount relatively to itself as Unit. Other-being, the Amount of Units, is Unit itself. Power is a multitude of Units each of which itself is this multitude. Quantum as indifferent determinate¬ ness varies ; but where this variation means raising to a power, this its other-being is limited purely by itself. — Thus in Power Quantum is posited as having returned upon itself: it is immediately itself and also its other-being.
The exponent of this Ratio no longer is an immediate Quantum as in Direct and also in Inverse Ratio. In the Ratio of Powers it is of a wholly qualitative nature ; it is the simple determinateness that Amount here is Unit and that Quantum in its other-being is self-identical. This also contains the aspect of its quantitative nature, which is, that limit or negation is not posited as existing immediately, but that Determinate Being is posited as continued into its otherness; for the true nature of quality is this, that it is quantity (or immediate determinateness in so far as it is transcended).
2. Ratio of Powers first appears as an external variation applied to any Quantum ; but it has this closer connexion with the concept of Quantum, that Quantum, in that Determinate Existence into which it has developed in this Ratio, reaches and completely realizes this concept : this Ratio represents what Qixantum is in itself, and expresses its determinateness or quality by which it is distinct from anything else. Quantum is determinateness indifferent and posited as transcended ; that is, determinateness as limit which also is no limit, and continues itself into its otherness, and consequently remains self-identical. It is thus posited in the Ratio of Powers, and its other-being or self-overpassing into another Quantum is determined by itself.
If we compare the progress of this realization in the Ratios with which we have dealt hitherto, then the quality of Quan¬ tum to be its own posited self-difference, is just this, that it is Ratio. As Direct Ratio it is, as such posited difference, only
general or immediate, so that its self-relation, which, as ex¬ ponent, it possesses in relation to its differences, counts only as the fixity of an Amount of Units. In Inverse Ratio, Quantum in its negative determination is an attitude of itself to itself, — to itself as its own negation, wherein, however, it has its value ; as affirmative self-relation it is an exponent, which, as Quantum, is the determinant of its moments only in itself. But in the Ratio of Powers it is present in the difference, because the difference is a self-difference. Externality of determinateness is the quality of Quantum; and according to its concept this externality is posited as its self-determination, as its self-relation and its Quality.
3. But Quantum, being now posited as it is in accordance with its concept, has passed over into another determination. This might also be expressed thus — that its determination is its determinateness, and its Being-in-Self its Determinate Being. It is Quantum in so far as the externality or indifference of determinateness (which means that it is that, as the phrase goes, which can be increased or diminished) counts and is posited only simply or immediately; and it has become its Other (namely Quality) in so far as this externality is now posited as mediated by Quantum and as being such a moment as refers itself in it to itself and is Being as Quality.
Thus first Quantity as such appears as opposed to Quality. But Quantity is itself a quality ; it is self-relating determinate¬ ness in general, distinct from that determinateness which is other to it, namely Quality as such. But it is not only a quality : the truth of Quality itself is Quantity, and the former has manifested itself as passing over into the latter. On the other hand, Quantity in its truth is externality which has returned to itself and is nqt indifferent. And thus it is Quality itself, and, outside this determination, Quality as such would be nothing. — In order that the totality may be posited, the twofold transition is required, — not only the transition of one determinateness into its other, but also the transition of the latter into the former, its regress into the first. The first tran¬ sition gives us only the identity in itself of both; — Quality is contained in Quantity, but so far Quantity is a one-sided determinateness. It is the result of the second transition that, conversely, Quantity is contained in Quality, and that it too
now for critique of point 3, that the content disseminated through the media shapes people’s worldview. here, i argue that it’s the form of the media that shapes people’s worldview
exists only as having been transcended. This is the regress into the first. This observation upon the necessity of the twofold transition is of great importance for the whole scientific method.
Now Quantum no longer is indifferent or external deter¬ mination, but, as such, is transcended, and is Quality, and is that by virtue of which something is what it is : this is the truth of Quantum, and this is Measure.
It has been explained above, in the Observations upon the Quantitative Infinite, that this and the difficulties which result have their origin in the qualitative moment which manifests itself in the quantitative; and further, how especially the qualitative moment of the Ratio of Powers runs into manifold developments and complexities. It was shown that the funda¬ mental flaw which prevented the concept from being appre¬ hended was, that a halt was made at the negative deter¬ mination of the infinite (where it is negation of Quantum), and that no progress was made to the simple affirmative determination which states that this is the qualitative. — Here it only remains to remark upon the intrusion, in philosophy, of quantitative forms into the pure qualitative forms of thought. The Ratio of Powers has especially been applied in recent times to conceptual determinations. The concept in its imme¬ diacy was called first power, in its other-being or difference (the Determinate Being of its moments) it was called second power, and in its return upon itself or totality, third power. — Here it is immediately evident that Power thus employed is a category essentially belonging to Quantum: this Power is not meant as potentia, the hvva/us of Aristotle. The Ratio of Powers thus expresses determinateness as reaching its truth in Difference, as Difference exists in the special notion of Quantum, but not as it exists in the Notion as such. Quantum does not yet by any means possess that negativity which belongs to the nature of the Notion, as posited in its peculiar determination : differences which are proper to Quantum are superficial determina¬ tions for the Notion itself, and are far from being determinate as they are determinate in the Notion. It was in the child¬ hood of philosophic thought that numbers were used (as by
Pythagoras) to designate universal and essential distinctions; and here first, second, or other Powers are in no way better than numbers. This was a rudimentary form of pure thinking comprehension ; the determinations of thought themselves were not discovered — that is, brought to consciousness for themselves — till after Pythagoras. To return from these to numerical determinations is the part of thought which knows its impotence and, in opposition to what our philosophic culture — no stranger to determinations of thought — had reached, commits the further folly of trying to vindicate this weakness as something new, superior, and in the line of progress.
In so far as Powers are used only as a symbolical expression, they are unobjectionable, just as much as are numbers and other symbols of concepts, — but also they are as objectionable as all symbolism whatever which attempts to represent pure conceptual or philosophic determinations. Philosophy needs no such help either from the sensible world or from active imagination or from subordinate fields of philosophical specu¬ lation, the determinations of which, therefore, are unfitted for higher spheres and for the whole. The latter happens whenever categories of the finite are applied to the infinite; the common determinations of force, or substantiality, cause and effect, and others, are themselves too only symbols used to express other relations, like vital and spiritual relations; that is, they are untrue determinations of those relations, and still more so are Powers of Quantum and numbered powers, both for such and for speculative relations generally. — If numbers, powers, the mathematical infinite, and the like are to be used not as symbols but as forms for philosophic deter¬ minations and hence themselves as philosophic forms, then first of all their philosophic meaning, that is, their conceptual determinateness, must be demonstrated. If this is done, they are superfluous designations: the conceptual determinateness designates itself, and its own is the only correct and fitting designation. The use of these forms is, therefore, nothing but a convenient means of escaping the trouble of seizing, pro¬ claiming, and justifying the conceptual determinations.
Abstractly the statement may be made that in Measure Quality and Quantity are united. Being as such is the imme¬ diate self-identity of determinateness. This immediacy of determinateness has transcended itself. Quantity is Being which has returned upon itself in such a manner that it is simple self-identity as indifference to determinateness. But this indifference is merely externality-possessing its determinateness not in itself but in an Other. Thirdly, there is self-relating externality : as self-relation it is also transcended externality and has in itself its own difference from itself. This difference as externality is the quantitative moment, and, as having returned to itself, the qualitative moment.
Modality is enumerated among the categories of transcendental idealism after Quantity and Quality, and where Relation is inserted ; so that it may be mentioned here. The category there means that it is the relation of the object to thought. In this idealism thought is essentially external to the thing-in-itself. In so far as other categories have only the transcendental determination of belonging to consciousness as its objective element, modality, which is the category of relation to the subject, to this extent relatively contains the determination of intro-reflection; that is, objectivity, which is stated to be a quality of the other categories, is lacking in the categories of modality; and these (in Kant’s expression) do not in the least add to the concept as determination of the object, but only express the relation to the possibility of cognition ( Critique of Pure Reason , 2nd Ed., see pp. 99, 266).— The categories which Kant groups under Modality — Possibility, Actuality, and Necessity — will occur later in their place; that infinitely important triple form was not applied by Kant— for with him it appeared only as a formal flash of light — to the genera of his categories (Quantity, Quality, etc.); and this name too he applied only to their species. It was impossible for
him, therefore, to find the third member for Quality and Quantity.
With Spinoza, too, the Mode is third after Substance and Attribute: he declares that the Mode is the affections of Sub¬ stance, or is that element in an Other by virtue of which it is comprehended. According to this concept this third element is only externality as such; and indeed it has been mentioned elsewhere that iii Spinoza in general rigid substantiality lacks the return into itself.
The observation here made extends more generally to those systems of Pantheism upon which thought has done its elabo¬ rating work. Being, the One, Substance, the Infinite, or Essence, is the first; in opposition to this abstract element the second, namely every kind of determinateness, can equally abstractly be grouped as that which is merely finite, accidental, perish¬ able, unessential, and non-essential; and this is the next and ordinary step in purely formal thought. But the connexion of the second with the first makes itself so evident that both must needs be taken as one unity; and thus with Spinoza the Attribute is the whole Substance, that is, as taken by Under¬ standing — itself a limitation or Mode : and Mode (which is the non-substantial in general which can be understood only from an Other) is thus the other extreme for Substance, the third in general. Indian Pantheism in all its monstrous imagination has also, taken abstractly, received this elaboration ; this is the tempering thread which leads through its riot to this point of moderate interest, that Brahma, the One of abstract thought, passes through the shape of Vishnu (especially in the form of Krishna) to the third form, Siva. The determination of this third is Mode, change, arising and passing away, the field of externality in general. If this Indian triad has led to a com¬ parison with the Christian, it must be recognized that they have a common element of conceptual determination; but it is essential that the difference be more definitely brought to consciousness: the difference is not only infinite, but true infinity constitutes the difference itself. According to its deter¬ mination this third principle is the explosion of substantial unity into its opposite, and not its return to itself, — the non¬ spiritual, not Spirit. In the true triad not only unity is found, but harmony — the consummation of a pregnant and real unity,
earlier, i gave the example that a content focused approach leads, for instance, to people engaging heavily with social media. so how does a form focused approach encourage us to engage with it?
which in its wholly concrete determination is Spirit. The principle of Mode and change does not indeed exclude unity : thus with Spinoza the Mode as such is the false, and Substance alone is the true, and everything must be reduced to it; which is a jettison of all content into the void, into a unity merely formal and without content; and similarly, Siva once more is the great whole, not distinct from Brahma: it is Brahma itself; that is, the difference and the determinateness vanish again, but are neither preserved nor transcended; unity is not led back to concrete unity, nor dissension to reconciliation. The highest goal for man transplanted into the sphere of arising and passing away, of modality in general, is submer¬ sion into unconsciousness, unity with Brahma, annihilation, which is the same as the Buddhist Nirvana, Nibbana, and so forth.
The Mode is abstract externality in general and indifference to qualitative and to quantitative determinations; and in essence what is external and unessential should not matter; but on the other hand it is often admitted that all depends upon the how and why. The Mode is thus declared essentially to belong to the substantial part of a thing. This very indefinite relation contains at least this, that this external part is the external not quite so abstractly. —
The ‘Mode here has the definite meaning of Measure. Spinoza’s Mode, like the Indian principle of Change, is the measureless. The Greek idea, though indeterminate as yet, that everything has a Measure (which led Parmenides to introduce, after abstract Leing, Necessity as the ancient Limit imposed on all things), is the beginning of a much higher concept than that contained in Substance and the difference between Mode and Substance.
When more fully developed and reflected, Measure becomes Necessity; Fate and Nemesis were generally limited to the determinateness of Measure, which meant that what is pre¬ sumptuous, or grows too high and great, is reduced to the other extreme by being brought down to annihilation, and that thus the mean of Measure, mediocrity, is restored. “The Absolute, or God, is the Measure of all things” is a definition not more strongly pantheistic but infinitely more true than “the Absolute, or God, is Bang.” Measure is
indeed an external way or manner, a more or less, but it is also reflected into itself, and is a determinateness not merely indifferent and external, but existing in itself. It is thus the concrete truth of Being ; and, therefore, mankind has revered in Measure something inviolable and holy.
The idea of Essence is already contained in Measure, namely that it is identical with itself in the immediacy of determinate- ness, so that this self-identity reduces the immediacy to a mediate; and also this mediate is mediated only through this externality, but is self-mediation; it is the reflection whose determinations are, but, thus being, exist only as moments of their negative unity. The qualitative is quantitative in Measure : determinateness or difference is indifferent, and, therefore, the difference is no difference, it is transcended: and this quanti- tativity, as return upon self, where it exists as the qualitative, constitutes that Being-in-and-for-Self which is Essence. But Measure is Essence only in itself or in the concept; and this concept of Measure has not yet been posited. Measure, while it is Measure, is the existent unity of the qualitative and the quantitative: its moments are as a Determinate Being, a Qpality and its Quanta, which are inseparable only in them¬ selves at first, but have not yet the meaning of this reflected determination. The development of Measure contains the differentiation of these moments, but also their relation, so that the identity which they are in themselves becomes their mutual relation, that is, is posited as such. The meaning of this development is the realization of Measure, where it posits itself as being related to itself and hence as a moment. This mediation determines it as being transcended; its immediacy and the immediacy of its moments disappear, and they exist as being reflected; and Measure, which thus has manifested itself as being what it is according to its concept, has passed over into Essence.
Measure now is immediate unity of the Qualitative and the Quantitative. Thus: —
first, it is a Quantum having qualitative meaning, and, therefore, existing as Measure. The process of further deter¬ mination is, that the difference of its moments (qualitative and quantitative determinateness) appears in this self-determined entity. These moments further determine themselves as each a
whole Measure, which thus is independent; and, these being essentially related to one another, Measure becomes, secondly, the relation of Specific Quanta as independent Measures. But at the same time their independence essentially is based on quantitative ratio and magnitudinal difference; thus their independence becomes a reciprocal transition. Consequently Measure perishes in the measureless. — But this Beyond of Measure is its negativity only in itself; hence, thirdly, the indifference of determinations of Measure is posited; and Measure (as real by virtue of the negativity which it contains) is posited as the Inverse Ratio of Measures, which, as independent Qualities, are essentially based only upon their Quantity and their negative relation to one another ; they thus turn out to be no more than moments of their truly independent unity, which is their intro-reflection and its positing, that is. Essence.
The development of Measure which has been attempted in what follows is exceedingly difficult ; it starts from immediate, external Measure, and should, therefore, on the one hand, proceed to the abstract further determination of the Quanti¬ tative (a natural mathematics), and, on the other, demonstrate the connexion between this Measure-determination and the Qualities of natural objects, at least in a general manner; for the exact demonstration of the connexion between Qualitative and Quantitative which arises from the concept of the concrete object belongs to the special science of the concrete, — examples of which may be looked up in the Encyclopaedia of the Philo¬ sophical Sciences, 3rd Ed. §§ 267 and 270, Observations on the Law of Gravity and that of free celestial motion. We may here observe generally that the various forms in which Measure realizes itself belong also to different spheres of natural reality. The complete and abstract indifference of developed Measure, that is, of its laws, can occur only in the sphere of Mechanism where the concrete corporeal is only matter, abstract itself; its qualitative differences essentially have the Quantitative for their determinateness; Space and Time are themselves pure externalities, and the multitude of materials or masses, the intensity of weight, are also external determinations having their peculiar determinateness in the Quantitative. However, such magnitudinal determinateness of the abstractly material
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is marred by plurality, and hence by a conflict of qualities, in the physical and even more in the organic sphere. And hare not merely the conflict of qualities as such operates, but Measure is subordinated to higher relations, and the immanent development of Measure is reduced to the simple form of Immediate Measure. The members of the animal organism have a Measure which, as a simple Quantum, is in a ratio to other Quanta of the other members : the proportions of the human body are fixed ratios of such Quanta; and natural science has much to discover of the connexion of such mag¬ nitudes with the organic functions, on which they wholly depend. But Motion is the nearest example of the reduction of an immanent Measure to a magnitude merely externally determined. In the celestial bodies it is free motion determined only by the concept, and, therefore, its magnitudes also depend on it alone (see above); but it is reduced from organic to arbitrary or mechanically regular motion, that is, to an alto¬ gether abstract, formal motion.
And in the realm of Spirit a peculiar and free development of Measure takes place even less. It is evident, for instance, that a republican constitution, like that of Athens, or an aristocratic constitution tempered by democracy, can exist only where the State has a certain magnitude, and that in developed civic society aggregates of individuals belonging to different trades are in a certain relation to one another; but this yields neither laws of Measure nor peculiar forms of it. In the realm of Spirit as such differences occur of intensity of character, strength of imagination, of feelings, of conceptions, and so on; but determination does not pass beyond this indeterminate concept of strength or weakness. And it is realized how poor and wholly void, in the end, are these so-called laws which are set up about the relation of strength and weakness of sensations, images, and so on, when the psychologies are examined which labour on these matters.
Qualitative Quantity is, firstly, an immediate specific Quantum; this,
secondly, is in a relation to another, and therefore becomes a quantitative specifying, a transcendence of indifferent Qjiantum. This Measure thus is a Rule, and contains, un- synthetized, the two moments of Measure, namely, self-existent quantitative determinateness and external Qjuantum. But being thus held apart the two sides become Qualities, and the Rule becomes a relation between them; hence Measure manifests itself,
thirdly, as a Relation of Qualities, which have first One Measure; later on, this specifies itself as an internal difference of Measure.
i. Measure is the simple self-relation of Quantum, its own determinateness determined by itself: thus Quantum exists qualitatively. And first, as immediate Measure, it is an imme¬ diate Quantum, determined accordingly as being just any Quantum; and the Quality belonging to it is similarly imme¬ diate and is determined similarly— Thus Quantum, as being this no longer indifferent limit, but a self-relating externality, is itself Quality ; and being differentiated from Quality it does not pass beyond it, nor does Quality pass beyond Quantum. It is thus determinateness which has returned into simple self-identity; it is one with Determinate Being as Determinate Being is one with its Quantum.
If the determination which has been reached is to be for¬ mulated, it may be expressed in the proposition that everything which exists has a Measure. All Determinate Being has a mag¬ nitude, and this magnitude belongs to the nature of Something itself; it constitutes its determinate nature and Being-in-Sdf.
Something is not indifferent to this magnitude, nor does it remain unchanged when the latter changes : a change in the magnitude would change its quality. As Measure, Quantum has ceased to be a limit which is no limit; it now is a deter¬ mination of the Thing in such a manner that an increase or decrease in this Quantum would destroy it. —
A Measure as standard in the ordinary sense is a Quantum which is arbitrarily taken as Unit, determinate by itself, against an external Amount. Such a unit can, of course, also be a unit determinate by itself in fact, like a foot and other original measures ; but, in so far as it is also used as standard for other things, it is for them an external and not their original Measure. — In this way the diameter of the earth or the length of the pendulum may be taken as specific Quantum for itself. But what fraction of the diameter of the earth or length of the pendulum is taken, and the degree of latitude under which the latter is taken when it is to be used as standard, — this is arbitrary. And for other things such a standard is still more something external. These have specified the general specific Quantum in yet another particular manner, and have thereby become particular things. It is, therefore, foolish to speak of a natural standard of things. Moreover, a general standard is designed to serve only for external comparison ; and in this most superficial meaning, where it is taken as General Measure, it is quite indifferent what is used as Measure. It is not meant to be a fundamental Measure, which would mean that in it the natural Measures of particular things would be represented and would hence, according to a Rule, be recognized as speci¬ fications of a universal Measure, the Measure of their universal body. But without this meaning an absolute standard is interesting and significant only as being common to all ; and such a common element is universal not in itself, but only by convention.
This immediate Measure is a simple magnitudinal deter¬ mination, like the magnitude (for example) of organic beings, of their members, etc. But everything that exists has a mag¬ nitude which makes it what it is and allows it to have Deter¬ minate Being. — As Quantum it is an indifferent magnitude, open to external determination and capable of oscillation along the scale of more and less. But, as Measure, it is also distinct
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from itself as Quantum (which is such an indifferent deter¬ mination), and is a limitation of this indifferent oscillation along a limit.
In Determinate Being quantitative determinateness thus is double; first, Quality is bound to it, but, secondly, oscillation may take place along it and Quality yet survive. It thus comes about that whatever has a Measure may perish when its Quantum is varied. In a manner this destruction appears unexpected in so far as changes can be made in Quantum while Measure and Quality are unchanged; and yet it can be made quite intelligible — for it is gradual. Recourse is so readily made to this category in order to render intelligible to the eye or to the mind the disappearance of a Quality or of something; for thus the illusion is created that one can almost be eye-witness of disappearance; for, Quantum being posited as limit external and variable by its very nature, change (as a change of Quantum only) needs no explanation. But in fact nothing is thereby explained; the change is also essentially the transition of one Quality into another, or (a more abstract transition) of one existence into a non-existence ; and this contains a determination different from that of “gradual,” which is only a decrease or increase and a one¬ sided retention of magnitude.
2. But already the ancients were aware of the connexion by which a change appearing as merely quantitative turns into one which is qualitative, and they illustrated the con¬ fusions which arise from ignorance of this connexion by popular examples ; the well-known eXeyxpi called those of the Bald and of the Heap belong here; these (according to Aristotle) are devices to compel the assertion of the contradictory of a first assertion. The question was, whether (for example) the re¬ moval of one hair from a head, or from a horse’s tail, produced baldness; or whether a heap ceased to be a heap after the removal of one grain. This could be denied without hesitation, for such a removal constitutes a wholly negligible quantitative difference; and thus one hair or grain is removed, and this is repeated, one only being removed after each acquiescence: there appears at last the qualitative change, — the head (or tail) is bare, the heap has vanished. At each acquiescence not only the repetition was forgotten, but also the fact that the
quantities negligible in themselves (like disbursements neg¬ ligible in themselves from a capital) add up, and that the sum forms the qualitative whole, which, therefore, at the end has vanished : the head is bald, and the purse empty.
The embarrassment and contradiction which result are no sophism in the ordinary meaning of the term : this contradic¬ tion is not vicious nor illusory. The error is committed by the Other which was assumed, namely by ordinary consciousness, when it takes a quantity as a merely indifferent limit, that is, precisely, in the definite meaning of a Quantity. This assump¬ tion is upset by the truth to which it is led, namely, that it is a moment of Measure and is connected with Quality : what is refitted is the one-sided clinging to abstract determinateness of Quantum. — And therefore these divagations are no idle and pedantic joke ; they are in themselves correct and the product of a consciousness which takes an interest in the phenomena which occur in thought.
Quantum when it is taken as indifferent limit is that side from which a Determinate Being can unsuspectedly be attacked and destroyed. It is the cunning of the Notion to seize it from this side, where its Quality does not appear to come into play; — and this so much so that the aggrandizement of a State or of a property, and so on, which leads in the end to disaster for the State or the owner of the property, may at first actually appear as their good fortune.
3. Measure in its immediacy is an ordinary Quality of a determinate magnitude belonging to it. Now that side accord¬ ing to which Quantum is indifferent limit along which oscilla¬ tion can take place while the Quality remains unchanged, is also different from its other side, according to which it is qualitative or specific. Both are magnitudinal determinations of one and the same thing; but, further, this difference must be taken as immediate in accordance with the immediacy in which Measure first is found : hence the two sides have different existences. The existence of Measure, then, which is magnitude determinate in itself, is, in its attitude to the existence of the variable and external side, a transcendence of its indifference; it is a specifying of Measure.
thirdly, both sides, as Qualities of specific quantitative determinateness, are related to one another as One Measure.
Rule, or Standard (which has already been spoken of), is first a magnitude determinate in itself; it is Unit with reference to a Quantum, which is a particular existence, existing in a Something other than the Something of the Rule, and measured by the Rule, that is, determined as Amount of that Unit which is the Rule. This comparison is an external activity, and the Unit is an arbitrary magnitude, which in turn can be posited as Amount (the foot as an Amount of inches). But Measure is not only external Rule, but being specific it must also in itself be related to an Other which is a Quantum.
Measure is a specific determining of external magnitude, that is, of the indifferent magnitude which is posited by some other existence in general in the Something of Measure ; Measure is itself a Qpantum, but, as distinguished from this, it is the qualitative element determining merely indifferent and external Quantum. Something has in it that side of Being-for-Other to which indifferent increase and decrease is proper. This immanent mensor is a Quality of Something, to which the same Quality in another Something is opposed; but in the latter the Quantum is relatively measureless as opposed to the former Quality which is determined as mensor.
In so far cs Something is a Measure in itself, a change in the magnitude of its Quality is external, and does not make of it an arithmetical plurality. But its Measure reacts against it, is in the relation of an intensive to this plurality, and assimilates it in a peculiar manner; it changes the change which is externally posited, makes this Quantum another, and,
by means of this Specification, manifests itself in this externality •as Being-for-Self. — The plurality which is specifically assimi¬ lated is itself a Quantum, dependent upon the other plurality, which also remains merely external to it. The specified plurality is consequently also variable; but it is not, therefore, a Quan¬ tum as such, but is external Quantum specified in a constant manner. Thus Measure has its determinate being as Ratio, and its specific element is in general the exponent of this Ratio.
It was seen, when these determinations were being con¬ sidered, that in Intensive and Extensive Quantum it is one and the same Quantum which we find, in one case in the form of intensity, and in the other in the form of extensity. In this difference the basic Quantum undergoes no change: the difference is only an external form. On the other hand, in Specifying Measure, Quantum in the first case exists in its immediate magnitude, but in the second is taken (by virtue of the exponent of the Ratio) in another Amount.
The exponent, which constitutes the specific element, might at first appear to be a fixed Quantum, as being the quotient of the ratio between the external and the qualitatively deter¬ minate term. But then it would be nothing but an external Quantum; by exponent nothing must here be understood but the qualitative moment itself which specifies the Quantum as such. The properly immanent qualitative element of Quantum is (as was seen above) only the Determination of Power. This it must be which constitutes the ratio, and which here, as the self-existent determination, has opposed Quantum as being external modification. The principle of Quantum is the numeri¬ cal One, which constitutes its determinateness-in-itself; and the relation of the numerical One is external; and the change (which is determined only by the nature of immediate Quantum as such) consists in itself in the addition of such a numerical One, of yet another, and so on. Thus external Quantum changes in arithmetical progression; and thus the specifying reaction of the qualitative nature of Measure produces another series, which is related to the first and increases and decreases with it, but in a ratio which is not determined by a numerical exponent, but is incommensurable with a number, according to a determination of powers.
Temperature— to cite an example— is a Quality where these two sides, of being external and specified Quantum, are dis¬ tinguished. As Quantum it is external temperature (even that of a body as the general medium), of which it is assumed that its change moves along the scale of arithmetical pro¬ gression, and that it increases or decreases uniformly ; whereas it is absorbed in a different manner by the different individual bodies which it includes, since these, by virtue of their immanent Measure, determine the temperature which they receive from without, and the change in the temperature of any one does not correspond directly with that of the medium or of any other. Different bodies, compared under the same temperature, give relative numbers of their specific heats, of their capacities for heat. But these capacities of bodies vary under different temperatures, and thus a change in the specific configuration is introduced. Hence a particular specification manifests itself in the increase or decrease of temperature. The relation of the temperature which is imagined as external, to the tem¬ perature of a given body, which also depends upon the former, has no fixed exponent of ratio; the increase or decrease of this heat does not progress uniformly with the increase or decrease of external heat. — A temperature is here assumed which is altogether external, and changes quite externally or purely quantitatively. But the temperature is itself the tem¬ perature of air or some other specific temperature. More closely considered, therefore, the ratio would really have to be taken not as the relation between a merely quantitative and a qualifying Quantum, but between two specific Quanta. And indeed Specifying Ratio will immediately proceed to determine itself in such a manner that the moments of Measure do not consist only of two sides of one and the same Quality, a quantitative side and a side qualifying the Quantum, but of the relation between two Qualities which in themselves are Measures.
i. The qualitative and self-determinate side of Quantum exists only as relation to the externally quantitative; as its
specification it is the transcendence of its externality, by virtue of which Quantum exists as such; thus it logically depends on, and begins from. Quantum. But Quantum differs from Quality itself also in a qualitative manner ; and this difference between the two must be posited in the immediacy of Being in general, which as yet is still the sphere of Measure : the two sides are thus qualitatively related to each other, and each for itself is such a Determinate Being ; and the one Quantum (as yet formal and not determinate in itself) is the Quantum of a Something and of its Quality, and also is the specific magnitude of these qualities, since it is the case that their mutual relation has proceeded to the determination of Measure in general. These Qualities are related to each other according to the determination of Measure, and this determination is their exponent; but they are so related in themselves already in the Being-for-Self of Measure: in its double existence Quantum is both external and specific, so that each of the different Quantities has this double determination in itself and also is absolutely interlocked with the other : it is just in this alone that the Qualities are determinate. Thus they are not only determinate existences existing for each other, but they are posited as inseparable, and the magnitudinal determinate¬ ness connected with them is a qualitative unity — one deter¬ mination of Measure in which, according to their concept, they cohere. Measure is thus the immanent quantitative mutual attitude of two Qualities.
2. In Measure the essential determination of variable mag¬ nitude appears, for Measure is Quantum as transcended, that is, no longer as that which it is supposed to be in order to be Quantum, but as Quantum and also something else; this Other is the Qualitative, and, as was determined, nothing else than its Ratio of Powers. This change is not yet posited in immediate Measure : any Quantum (and in fact one individual Quantum) is there connected with the one Quality. But in the Specifying of Measure (the preceding determination) as a changing of merely external Quantum by means of the Qualitative, a difference is posited between the two magni¬ tudinal determinatenesses, and thus a plurality of Measures in a common external Quantum; and Quantum shows itself as existing Measure only here where it is differentiated from
itself, for here it is manifested both as one and the (e.g. the same temperature of the medium) and also as different (quantitative) Determinate Being (in the different temperatures of the bodies contained in the medium). This Hiffprpntjarin^ of Quantum in the different Qualities (the different bodies) gives another form of Measure, namely the one where both sides are mutually related as Quanta qualitatively determined : this might be called Realized Measure.
Magnitude as magnitude in general is variable, for its determinateness is a limit which also is no limit; variation then affects only a particular Quantum, for which another is substituted. But the true change is the change of Quantum as such; this leads to the determination, interesting if taken in this way, of the variable magnitude of higher mathematics: here no halt must be made at the mere form of variability in general, nor must any other determination be introduced than the simple determination of the concept, according to which the Other of Quantum is only the Qualitative. Thus the true determination of real variable magnitude is that it is magnitude qualitatively determined, that is (as has sufficiently been shown) determined by a Ratio of Powers : in this variable magnitude it is posited that Quantum is counted not as such, but according to its other, or qualitative, determination.