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.
a) (x + h)³ = x³ + 3hx² + 3h²x + h³ ;
b) (x + h)³ - x³ = 3x²h + 3xh² + h³ ;
c) ((x + h)³ - x³)/h = 3x² + 3xh + h² .
If h becomes = 0, then
((x + 0)³ - x³)/0 or (x³ - x³)/0 = 0/0 or dy/dx
and the right-hand side = 3x², so that
dy/dx = 3x² .
y = x³; y1 = x1³ ;
y1 - y = x1³ - x³ = (x1 - x) (x1² + x1x + x²) ;
y1 - y)/(x1 - x) or = dy/dx = x² + xx + x² ;
dy/dx = 3x² .
II) Let us set x1 - x = h. Then:
1) (x1 - x) (x1² + x1x + x²) = h(x1² + x1x + x²) ;
2) so that:
(y1 - y)/h = x1² + x1x + x² .
In 1) the coefficient of h is not the completed derivative, like f’ above, but rather f¹; the division of both sides by h, therefore, also leads not to dy/dx, but rather
Δy/h or Δy/Δx = x1² + x1x + x²
If we begin on the other side in I c), namely in
(f(x + h) - f(x))/h or (y1 - y)/h = 3x² + 3xh + h² ,
from the assumption that the more the value of h decreases on the right-hand side, so much the more does the value of the terms 3h + h² decrease,88 so that the value as well of the entire right-hand side 3x² + 3xh + h² more and more closely approaches the value 3x², we the must set down, however : ‘yet without being able to coincide with it’.
3x² thus becomes a value which the series constantly approaches, without ever reaching it, and thus, even more, without ever being able to exceed it. In this sense 3x² becomes the limit value89 of the series 3x² + 3xh + h².
On the other side the quantity (y1 - y)/h (or (y1 - y)/(x1 - x)) also decreases all the more, the more its denominator h decreases.90 Since, however, (y1 - y)/h is the equivalent of 3x² + 3xh + h² the limit value of the series is also the ratio’s own limit value in the same sens that it is the limit value of the equivalents series.
However, as soon as we set h = 0, the terms on the right-hand side vanish, making 3x² the limit of its value; now 3x² is the first derivative of x³ and so = f’(x). As f’(x) it indicates that an f’’(x) is also derivable from it (in the given case it = 6x) etc., and thus that the increment f’(x) or 3x² is not = the sum of the increments which can be developed from f(x) = x³. Were f(x) itself an infinite series, so naturally the series of increments which can be developed from it would be infinite as well. In this sense, however, the developed series of increments becomes, as soon as I break it off, the limit value of the development, where limit value here is in the usual algebraic or arithmetic meaning, just as the developed part of an endless decimal fraction becomes the limit of its possible development, a limit which is satisfactory on practical or theoretical grounds. This has absolutely nothing in common with the limit value in the first sense.
Here in the second sense the limit value may be arbitrarily increased, while there it may be only decreased. Furthermore
(y1 - y)/h = (y1 - y)/(x1 - x) ,
so long as h is only decreased, only approaches the expression 0/0; this is a limit which it may never attain and still less exceed, and thus far 0/0 may be considered its limit value.91
As soon, however, as (y1 - y)/h is transformed to 0/0 = dy/dx, the latter has ceased to be the limit value of (y1 - y)/h, since the latter has itself disappeared into its limit.92 With respect to its earlier form, (y1 - y)/h or (y1 - y)/(x1 - x), we may only say that 0/0 is its absolute minimal expression which, treated in isolation, is no expression of value (Wertausdruck); but 0/0 (or dy/dx) now has 3x² opposite it as its real equivalent, that is f’(x).
And so in the equation
0/0 ( or dy/dx) = f’(x)
neither of the two sides is the limit value of the other. They do not have a limit relationship (Grenzverhältnis) to one another, but rather a relationship of equivalence (Aquivalentverhältnis). If I have 6/3 = 2 then neither is 2 the limit of 6/3 nor is 6/3 the limit of 2. This simply comes from the well-worn tautology that the value of a quantity = the limit of its value.
The concept of the limit value may therefore be interpreted wrongly, and is constantly interpreted wrongly (missdeutet). It is applied in differential equations93 as a means of preparing the way for setting x1 - x or h = 0 and of bringing the latter closer to its presentation: - a childishness which has its origin in the first mystical and mystifying methods of calculus.
In the application of differential equations to curves, etc., it really serves to make things more apparent geometrically.
In the manuscript ‘On the History of Differential Calculus’ Marx notes that from the simple difference in the form of representation of the change in the value of the function originate essential differences in the treatment of differential calculus (see p.102). Regarding this he made reference to the ‘introductory pages’ in which he developed this thought ‘in the analysis of d’Alembert’s method’ (see ibid.) These sheets are of two groups: sheets of one group are marked with the capital Latin letters A to H (see p.471 [Yanovskaya, 1968]), and sheets of the other group with small Latin letters from a to n (see p.498 [Yanovskaya, 1968]).
Since d’Alembert defines the derivative by means of the concept of limit, Marx naturally begins his analysis of the method with a critique of the concept of limit, the inadequacy of which is made clear with the material presented in Appendix I (see ‘Concerning the Concept of “Limit” in the Sources Consulted by Marx’, p.153). This part of the manuscript occupies sheets A to D (published under the title corresponding to its contents, ‘On the Ambiguity of the Terms “Limit” and “Limit Value” ‘). Also directly related to the above-mentioned passage in the manuscript on the history of differential calculus are sheets E to H, published here under the title, ‘Comparison of d’Alembert’s Method to the Algebraic Method’. And devoted to essentially the same question are sheets a to g of the other group, which are published here under the title ‘Analysis of d’Alembert’s Method by Means of yet Another Example.’ (For the contents of the remaining sheets of this group see pp.468-470 [Yanovskaya, 1968].) In conformity with Marx’s reference to the appended separate sheets devoted to the analysis of d’Alembert’s method, they are grouped together here under the general title, ‘Appendices to the Manuscript “On the History of Differential Calculus”: Analysis of d’Alembert’s Method’ (pp121-132).
In other words, it is proposed to consider here the expression 3x² + 3xh + h² for non-negative values of x and h under the assumption that h tends unboundedly towards zero, remaining different from zero. We recall that in the sources which Marx used there was as yet no concept of absolute value, so that he was not required to consider the sum of all non-negative terms.
89 Here Marx comes to the basis for his later conclusion, that ‘the concept of the limit value may be interpreted wrongly, and in constantly interpreted wrongly’ (see p.126), as a consequence of which it is appropriate to replace it by some new term which is unambiguously understandable. As such he proposes the term ‘absolute minimal expression’, by which is meant the limit in the usual present-day meaning of the word (see p.126 and Appendix I, p.143). Marx’s criticism of the ‘limit value’ defined here and of the way this concept is used in Hind’s and Boucharlat’s textbook refers first of all to the fact that the ‘limit’ is considered there as actual; that is, it is regarded as ‘the last’ value of the function for ‘the last’ value of the argument, and therefore represents ‘a childishness which has its origin in the first mystical and mystifying method of calculus’ (see p.126). In this particular paragraph he obviously has in mind the ‘limit value’ in the meaning of the definition introduced by Hind (see Appendix I, p.145), who in practice treats it as coinciding with the one-sided limit of a function where the argument approaches a certain number from the right or from the left: in the given case, with the one-sided limit from the right of the function 3x² + 3xh + h², considered as a function of h as h→+0. In contrast to Hind, however, Marx emphasizes that this ‘limit value’ only has meaning if it is not understood as taking place but is calculated with the condition that h ≠ 0 (here h > 0); that is, he treats it exactly as we do today. At the same time the application of this to the function in consideration, 3x² + 3xh + h², does not violate the requirement contained in the definition of ‘limit value’ (as the exact upper or lower bound to the value of the variable) with which Hind’s textbook begins. In fact, as Marx notes, this function firstly, as h approaches zero, constantly approaches its own limit (the lower one clearly), and secondly, consequently all the more never passes beyond it; that is, it explicitly satisfies both conditions of Hind’s definition (Hind himself usually did not verify the satisfaction of these requirements; see Appendix I, p.145).
90 If the (one-sided) limit of the function 3x² + 3xh + h² at the approach of h to zero (from the right; that is, as h decreases) is interpreted actually, that is, the argument h is supposed to attain its limit (‘last’) value 0, then from the multiplicity of values of the function with respect to which, according to Hind’s definition, the limit must be the exact lower bound, it is sufficient to choose the set consisting of only the one value of the function at h = 0 (see Appendix I, p.145), in the given case consequently of only one number 3x² - which, however, as Marx says below, it would be a ‘well-worn tautology’ to regard as the limit value for 3x² as h approaches zero. In other words, to speak naturally of 3x² as the limit value of 3x² + 3xh + h² as h approaches zero at the same time as regarding 3x² as the limit value of 3x² itself as h approaches zero is not intelligible here - most of all because it is in general superfluous: it gives us nothing new.
91 This expression 0/0 is considered here to be the limit of the quotient (y1 - y)/(x1 - x), as was done similarly in Boucharlat’s textbook (see Appendix I, p.149), but with the difference that here the limit value (here again in Hind’s sense) of the functions x1 - x and y1 - y as x1→+x is not understood by Marx in an actual sense, that is, it remains an assumption that x1 ≠ x (here x1 > x).
92 Here again reference is made to the fact that 0/0 (or dy/dx) is impossible to interpret actually, that is, as the value of the ratio y1 - y/h at h = 0, since in that case, following Hind and obtaining the limiting expression 0/0 by simply supposing h = 0, one would have to admit that the consideration of this expression, in which no trace remains of the ratio (y1 - y)/h which contained the variable h, as the limiting value for the same 0/0 (regarded as the ‘constant’ function of h) as h→+0, in general gives no new result. However, for the expression (y1 - y)/h when considered for h distinct from zero (here h > 0), it is precisely 0/0 which, standing opposite the derived function ‘as its real equivalent’, is, as Marx says, ‘its absolute minimal expression’, that is, the limit in the usual present-day sense.
93 The original had initially: ‘applied in the above differential equations’ (auf obige Differentialgleichungen), but Marx crossed out the word ‘obige’. It is however clear that here as previously, this does not concern equations in the proper sense of the word, but rather the fundamental formulae of differential calculus having the form of equalities.