Marx's Mathematical Manuscripts 1881
Written: August, 1881;
Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Russian translation, in Pod znamenem marksizma, 1933.
1) Mystical Differential Calculus. x1 = x + Δx from the beginning changes into x1 = x + dx or x + x., where dx is assumed by metaphysical explanation. First it exists, and then it is explained.
Then, however, y1 = y + dy or y1 = y + y.. From the arbitrary assumption the consequence follows that in the expansion of the binomial x + Δx or x + x., the terms in x and Δx which are obtained in addition to the first derivative, for instance, must be juggled away in order to obtain the correct result etc. etc. Since the real foundation of the differential calculus proceeds from this last result, namely from the differentials which anticipate and are not derived but instead are assumed by explanation, then dy/dx or y./x. as well, the symbolic differential coefficient, is anticipated by this explanation.
If the increment of x = Δx and the increment of the variable dependent on it = Δy, then it is self-evident (versteht sich von selbst) that Δy/Δx represents the ratio of the increments of x and y. This implies, however, that Δx figures in the denominator, that is the increase of the independent variable is in the denominator instead of the numerator, not the reverse; while the final result of the development of the differential form, namely the differential, is also given in the very beginning by the assumed differentials.*
If I assume the simplest possible (allereinfachste) ratio of the dependent variable y to the independent variable x, then y = x. Then I know that dy = dx or y. = x.. Since, however, I seek the derivative of the independent [variable] x, which here = x., I therefore have to divide64 both sides by x. or dx, so that:
dy/dx or y./x. = 1.
I therefore know once and for all that in the symbolic differential coefficient the increment [of the independent variable] must be placed in the denominator and not in the numerator.
Beginning, however, with functions of x in the second degree, the derivative is found immediately by means of the binomial theorem [which provides an expansion] where it appears ready made (fix und fertig) in the second term combined with dx or x.; that is with the increment of the first degree + the terms to be juggled away. The sleight of hand (Eskamotage), however, is unwittingly mathematically correct, because it only juggles away errors of calculation arising from the original sleight-of-hand in the very beginning.
x1 = x + Δx is to be changed to
x1 = x + dx or x + x. ,
whence this differential binomial may then be treated as are the usual binomials, which from the technical standpoint would be very convenient.
The only question which still could be raised: why the mysterious suppression of the terms standing in the way? That specifically assumes that one knows they stand in the way and do not truly belong to the derivative.
The answer is very simple: this is found purely by experiment. Not only have the true derivatives been known for a long time, both of many more complicated functions of x as well as of their analytic forms as equations of curves, etc., but they have also been discovered by means of the most decisive experiment possible, namely by the treatment of the simplest algebraic function of second degree for example:
y = x²
y + dy = (x + dx)² = x² + 2xdx + dx² ,
y + y. = ( x + x.)² = x² + 2xx. +x.² .
If we subtract the original function, x²(y = x²), from both sides, then:
dy = 2xdx + dx²
y. = 2xx. + x.x. ;
I suppress the last terms on both [right] sides, then:
dy = 2xdx, y. = 2xx. ,
dy/dx = 2x ,
y./x. = 2x .
We know, however, that the first term out of (x + a)² is x² ; the second 2xa ; if I divide this expression by a, as above 2xdx by dx or 2xx. by x., we then obtain 2x as the first derivative of x², namely the increase in x,65 which the binomial has added to x². Therefore the dx² or x.x. had to be suppressed in order to find the derivative; completely neglecting the fact that nothing could begin with dx² or x.x.*2 in themselves.
In the experimental method, therefore, once comes - right at the second step - necessarily to the insight that dx² or x.x. has to be juggled away, not only to obtain the true result but any result at all.
Secondly, however, we had in
2xdx + dx² or 2xx. + x.x.
the true mathematical expression (second and third terms) of the binomial (x + dx)² or (x + x.)². That this mathematically correct result rests on the mathematically basically false assumption that x1 - x = Δx is from the beginning x1 - x = dx or x., was not known.66
In other words, instead of using sleight of hand, one obtained the same result by means of an algebraic operation of the simplest kind and presented it to the mathematical world.
Therefore: mathematicians (man ... selbst) really believed in the mysterious character of the newly-discovered means of calculation which led to the correct (and, particularly in the geometric application, surprising) result by means of a positively false mathematical procedure. In this manner they became themselves mystified, rated the new discovery all the more highly, enraged all the more greatly the crowd of old orthodox mathematicians, and elicited the shrieks of hostility which echoed even in the world of non-specialists and which were necessary for the blazing of this new path.
2) Rational Differential Calculus. D’Alembert starts directly from the point de départ (sic) of Newton and Leibnitz: x1 = x + dx. But the immediately makes the fundamental correction: x1 = x + Δx, that is, x and an undefined but prima facie finite increment which he calls h. The transformation of this h or Δx into dx (he uses the Leibnitz notation, like all Frenchmen) is first found as the final result of the development or at least just before the gate swings shut (vor Toresschluss), while in the mystics and the initiators of the calculus it appears as the starting point (d’Alembert himself begins with the symbolic side,*3 but first transforms it symbolically). By this means he immediately succeeds in two ways.67
a) The ratio of differences
(f(x + h) - f(x))/h = (f(x + h) - f(x))/(x1 - x)
s the starting point of his construction (Bildung).
1) [the difference] f(x + h) - f(x), corresponding to the given algebraic function in x, stands out as soon as you replace x itself with its increment x + h in the original function in x, for example, in x³. This form ( = y1 - y, if y = f(x)) is that of the difference of the function, whose transformation into a ratio of the increment of the function to the increment of the independent variable now requires a development, so that it plays a real role instead of a merely nominal one, as it does with the mystics; for, if I have in these authors
f(x) = x³ ,
f(x + h) = (x + h)³ = x³ + 3x²h + 3xh² + h³ ,
then I know from the very beginning, that in
f(x + h) - f(x) = x³ + 3x²h + 3xh² + h³ - x³ ,
the opposing sides are to be reduced to the increment. This needn’t even be written down, since I see that on the second side the increment of x³ = the three following terms as well as that in f(x + h) - f(x), only the increment of f(x) remains, or dy. The first difference equation (Differenzgleichung) therefore only plays a role which from the very beginning is to disappear again. The increments stand opposite one another on both sides, and if I have them then I have from the definition of dx, dy that dy/dx or y./x. Is the ratio etc.; I therefore do not need the first difference, formed by the subtraction of the original function in x from the altered (by the replacement of x by x + h) function (the increased function), in order to construct dy/dx or y./x..
In d’Alembert it is necessary to hold fast to this difference because the steps of the development (Entwicklungsbewegungen) are to be executed upon it. In place of the positive expression of the difference, namely the increment, the negative expression of the increment, namely the difference, and thus f(x + h) - f(x), therefore comes to the fore on the left-hand side. And this emphasis on the difference instead of the increment (‘fluxion’ in Newton) is foreshadowed at least in the dy of Leibnitzian notation as opposed to the Newtonian y..
2) f(x + h) - f(x) = 3x²h + 3xh² + h³.
When both sides have been divided by h, we obtain
(f(x + h) - f(x))/h = 3x² + 3xh + h² .
Thereby is formed on the left-hand side
(f(x + h) - f(x))/h = ((x + h) - f(x))/(x1 - x)
which therefore appears as a derived ratio of finite differences, while with the mystics it was a completed ratio of increments given by the definitions of dx or x. and dy or y..
3) Now when in
(f(x + h) - f(x))/h = (f(x + h) - f(x))/(x1 - x)
h is set = 0, or x1 = x so that x1 - x = 0, this expression is transformed to dy/dx, while by means of this setting h = 0 the terms 3xh + h² all become [zero] simultaneously, and this by means of a correct mathematical operation. They are thus now discarded without sleight of hand. One obtains:
4) 0/0 or dy/dx = 3x² = f’(x) .
Just as with the mystics, this already existed as given, as soon as x became x + h, for (x + h)³ in place of x³ produces x³ + 3x²h + etc., where 3x² already appears in the second term of the series as the coefficient of h to the first power. The derivation is therefore essentially [the] same as in Leibnitz and Newton, but the ready-made derivative 3x² is separated in a strictly algebraic manner from its other companions. It is no development but rather a separation of the f’(x) - here 3x² - from its factor h and from the neighbouring terms marching in closed ranks in the series. What has on the other hand really been developed is the left-hand, symbolic side, namely dx, dy, and their ratio, the symbolic differential coefficient dy/dx = 0/0 (rather the inverse 0/0 = dy/dx), which in turn once more generates certain metaphysical shudderings, although the symbol has been mathematically derived.
D’Alembert stripped the mystical veil from the differential calculus and took an enormous step forward. Although his Traité des fluides appeared in 1744 (see p.15*4), the Leibnitzian method continued to prevail for years in France. It is hardly necessary to remark that Newton prevailed in England until the first decades of the 19th century. But here as in France earlier d’Alembert’s foundation has been dominant until today, with some modifications.
3) Purely Algebraic Differential Calculus. Lagrange, ‘Théorie des fonctions analytiques’ (1797 and 1813). Just as under I) and 2), the first starting point is the increased x; if
y or f(x) = etc.,
then it is y1 or f(x + dx) in the mystical method, y1 or f(x + h) (= f(x + Δx)) in the rational one. This binomial starting point immediately produces the binomial expansion on the other† side, for example:
xm + mxm-1h + etc.,
where the second term mxm-1h already yields ready-made the real differential coefficient sought, mxm-1 .
a) When x + h replaces x in a given original function of x, f(x + h) is related to the series expansion (Entwicklungsreihe) opposite it in exactly the same way that the undeveloped general expression in algebra, in particular the binomial, is related to its corresponding series expansion, as (x + h)³, for example in
(x + h)³ = x³ + 3x²h + etc.,
is related to its equivalent series expansion x³ + 3x²h + etc. With that step f(x + h) enters into the very same algebraic relationship (only using variable quantities) which the general expression has toward its expansion throughout algebra, the relationship, for example, which a/(a - x) in
a/(a - x) = 1 + x/a + x²/a² + x³/a³ + etc.,
has toward the series expansion 1 + etc., or which sin(x + h) in
sin(x + h) = sin x cos h + cos x sin h
has toward the expansion standing opposite it.
D’Alembert merely algebraicised (x + dx) or (x + x.) into (x + h), and thus f(x + h) from y + dy, y + y. into f(x + h). But Lagrange reduces the entire expression (Gesamtausdruck) to a purely algebraic character, since he places it, as a general underdeveloped expression, opposite the series expansion to be derived from it.
b) In the first method 1), as well as the rational one 2), the real coefficient sought is is fabricated ready-made by means of the binomial theorem; it is found at once in the second term of the series expansion, the term which therefore is necessarily combined with h¹. All the rest of the differential process then, whether in 1) or in 2), is a luxury. We therefore throw the needless ballast overboard. From the binomial expansion we know once and for all that the first real coefficient is the factor of h , the second that of h², and so on. The real differential coefficients are nothing other than those of the binomially developed series of derived functions of the original function in x (and the introduction of this category of derived function one of the most important). As for the separate differential forms, we know that Δx is transformed into dx, Δy into dy, and that the symbolic figure of dy/dx represents the first derivative, the symbolic figure d²y/dx² represents the second derivative, the coefficient of (1/2)·h², etc. We may thus allow the symmetry of half of our purely algebraically obtained result to appear at the same time in these its differential equivalent quantities (Differentialäquivalenten) - a matter of nomenclature alone, all that remains from differential calculus proper. The whole problem is then resolved into finding (algebraic) methods ‘of developing all kinds of functions of x + h in integral ascending powers of h, which in many cases cannot be effected without great prolixity of operation’.68
Until this point there is nothing in Lagrange which could not be a direct result of d’Alembert’s method (since this includes also the entire development of the mystics, only corrected).
c) While the development, therefore, of y1 or f(x + h) = etc. steps into the place of the differential calculus up to now [and thereby, in fact, clarifies the mystery of the methods proceeding form
y + dy or y + y., x + dx or x + x. ,
namely that their real development rests on the application of the binomial theorem, while they represent from the very beginning the increased x1 as x + dx, the increased y1 as y + dy, and thus transform a monomial into a binomial], the task now becomes, since we have in f(x + h) a function without degree before us, the general undeveloped expression itself only, to derive algebraically from this undeveloped expression the general, and therefore valid for all power functions of x, series expansion.
Here Lagrange takes as his immediate starting point for the algebraicisation of the differential calculus the theorem of Taylor outlived by Newton and the Newtonians69 which in fact is the most general, comprehensive theorem and at the same time operational formula of differential calculus, namely the series expansion, expressed in symbolic differential coefficients, of y1 or f(x + h), viz:
y1 or f(x + h)
= y ( or f(x)) + (dy/dx)·h + (d²y/dx²)·h²/ + (d³y/dx³)·h³/[2·3] + (d⁴y/dx⁴)·h⁴/[2·3·4] + etc.
d) Investigation of Taylor’s and MacLaurin’s theorem to be added here.70
e) Lagrange’s algebraic expansion of f(x + h) into a equivalent series, which Taylor’s dy/dx etc. replaces, and it may only still be the symbolic differential expression of the algebraically derived functions of x. (This is to be continued from here on.71)
Marx distinguishes the differentials (die Differentiellen) dx and dy, the infinitesimals of the differences Δx and Δy, from the differential (das Differential) : dy = f’(x)dx. - Trans.
64 If y. = x. and y itself is x, then in order to obtain an equality in which one side does contain the differential symbol x. it is sufficient simply to divide both sides of the equality y. = x. by x..
65 ‘Zuwachs in x’ (‘increase in x’) obviously signifies here a new function in x obtained from the initial function x² - in addition to it, so to speak - by means of the binomial theorem: as the coefficient dx in the expansion of (x + dx)².
*2 Printed edition has misprint xx. here. - Trans.
66 Obviously this refers to the fact that the immediate result of the application of the binomial is dy = 2xdx + dx², not dy = 2xdx. But the former equality appears to be mathematically correct only as a result of an incorrect premise.
*3 Traditionally the left-hand side - Trans.
67 The meaning of the expression ‘succeeds in two ways’ remains obscure. After the colon there follows a point a) without a point b). Perhaps the ‘two ways’ here are composed of first, the fact that on the left-hand side the fraction Δy/Δx is transformed into dy/dx (and not identified from the very beginning with dy/dx), and second, the fact that on the right-hand side the terms 3xh + h² are now obtained by means of correct mathematical operations and not by using some sleight of hand.
*4 See p.76
† i.e. right-hand - Trans.
68 The expression in quotation marks has been copied from Hind’s textbook cited above (§99, pp.128-129).
69 He obviously has in mind that Taylor’s theorem was published in his collection Methodus incrementorum in 1715, that is, during the life of Newton, in whose works this theorem does not appear. See in addition Appendix VI p.182.
70 For material related to the theorems of MacLaurin and Taylor, see pp.109-119 [this edition], 412, 441, 493, 498 [Yanovskaya, 1968].
71 For Marx’s exposition and critique of the fundamental ideas of Lagrange’s theory of analytic functions, see p.113 of this edition.