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.
As soon as we reach the differentiation of f(u,z)[=uz], where the variables u and z are both functions of x, we obtain - in contrast to the earlier cases which had only one dependent variable, namely, y - differential expressions on both sides, as follows:
in the first instance
dy/dx = z⋅du/dx + u⋅dz/dx ;
in the second, reduced form
dy = z⋅du + u⋅dz ,
which last also has a form different from that in one dependent variable, as for example, dy = maxm-1dx, since here that immediately gives us the dy/dx relieved of differential symbols f’(x) = maxm-1, which is by no means the case in dy = z⋅du + u⋅dz. The equations with one dependent variable showed us once and for all how the derived functions of [functions in] x, in this case of xm, were obtained through actual differentiation [taking of differences] and their later cancellation, and at the same time how there arose the symbolic equivalent 0/0 = dy/dx for the derived function. The substitution 0/0 = dy/dx here appears not only permissible but even necessary, since 0/0 in its primitive (waldursprünglichen) form = any quantity, because 0/0 = X always leads to 0 = 0. 0/0 appears here, however, equal to an exactly defined (ganz bestimmtem) specific value, = mxm-1, and it itself the symbolic result of the operations whereby this value is derived from xm; it is expressed as such a result in dy/dx. Thus dy/dx (= 0/0) is established from its origin as the symbolic value of differential expression of the already derived f’(x), not, conversely, f’(x) obtained by means of the symbol dy/dx.
At the same time, however, as soon as we have achieved this result and we therefore already operate on the ground (Boden) of differential calculus, we can reverse [the process]; if, for example, we have
xm = f(x) = y
to differentiate, we know immediately (von vornherein)
dy = mxm-1dx
or
dy/dx = mxm-1.
Thus here we begin with the symbol; it no longer figures as the result of a derivation from the function [of] x; rather instead as already a symbolic expression35 which indicates which operations to perform upon f(x) in order to obtain the real value of dy/dx, i.e. f’(x). In the first case 0/0 or dy/dx is obtained as the symbolic equivalent of f’(x); and this is necessarily first, in order to reveal the origin of dy/dx; in the second case f’(x) is obtained as the real value of the symbol dy/dx. But then, where the symbols dy/dx, d²y/dx² become the operational formulae (Operationsformeln) of differential calculus,36 they may as such formulae also appear on the right-hand side of the equation, as was already the case in the simplest example dy = f’(x)dx. If such an equation in its final form does not immediately give us, as in this case, dy/dx = f’(x), etc., then this is proof that it is an equation which simply expresses symbolically which operations are to be performed in application to defined (bestimmtem) functions.
And this is the case - and the simplest possible case - immediately in d(uz), where u and z are both variables while both are also functions of the same third variable, i.e. of x.37
Given to be differentiated f(x) or y = uz, where u and z are both variables dependent on x. Then
y1 = u1z1
and
y1 - y = u1z1 - uz .
Thus:
(y1 - y)/(x1 - x) = u1z1/(x1 - x) - uz/(x1 - x)
or
Δy/Δx = (u1z1 - uz)/(x1 - x) .
But
u1z1 - uz = z1(u1 - u) + u(z1 - z) ,
since this is equivalent to
z1u1 - z1u + uz1 - uz = z1u1 - uz .
Therefore:
(u1z1 - uz)/(x1 - x) = z1⋅(u1 - u)/(x1 - x) + u⋅(z1 - z)/(x1 - x) .
If now on both sides x1 - x becomes = 0, or x1 = x, then we would have u1 - u = 0, so that u1 = u, and z1 - z = 0, so that z1 = z; we therefore obtain
dy/dx = z⋅du/dx + u⋅dz/dx
and therefore
d(uz) or dy = z⋅du + u⋅dz .
At this point one may note in this differentiation of uz - in distinction to our earlier cases, where we had only one dependent variable tat here we immediately find differential symbols on both sides of the equation, namely:
in the first instance
dy/dx = z⋅du/dx + u⋅dz/dx ;
in the second
d(uz) or dy = z⋅du + u⋅dz
which also has a different form from that with one independent variable, such as for example, dy = f’(x)dx; for here division by dx immediately gives us dy/dx = f’(x)dx which contains specific value (Spezialwert) free of symbolic coefficients, derived from any function of x, f’(x): which is in no sense the case in dy = z⋅du + u⋅dz.
It has been shown how, in functions with only one independent variable, from one function of x, for example f(x) = xm, a second function of x, f’(x), or, in the given case mxm-1 may be derived by means of actual differentiation and subsequent cancellation alone, and at the same time how from this process the symbolic equivalent 0/0 = dy/dx for the derived function originates on the left-hand side of the equation.
Further: the substitution 0/0 = dy/dx here was not only permissible but mathematically necessary. Since 0/0 in its own primitive form may have any magnitude at all, for 0/0 = X always gives 0 = 0. Here, however, 0/0 appears as the symbolic equivalent of a completely defined real value, as above, for example, mxm-1, and is itself only the result of the operations whereby this value was derived from xm; as such a result it is firmly fixed (festgehalten) in the form dy/dx.
Here, therefore, where dy/dx ( = 0/0 ) is established in its origin, f’(x) is by no means found by using the symbol dy/dx; rather instead of the already derived function of x.
Once we have obtained this result, however, we can proceed in reverse. Given an f(x), e.g. xm, to differentiate, we then first look for the value of dy and find dy = mxm-1dx, so that dy/dx = mxm-1. Here the symbolic expression appears (figuriert) as the point of departure. [We] are thus (so) already operating on the ground of differential calculus; that is, dy/dx etc. already perform as formulae which indicate which known differential operations to apply to the function of x. In the first case dy/dx ( = 0/0 ) was obtained as the symbolic equivalent of f’(x), in the second f’(x) was sought and obtained as the real value of the symbols dy/dx , d²y/dx² , etc.
These symbols having already served as operational formulae (Operationsformeln) of differential calculus, they may then also appear on the right-hand side of the equation, as already happened in the simplest case, dy = f’(x)dx. If such an equation in its final form is not immediately reducible, as in the case mentioned, to dy/dx = f’(x), that is to a real value, then that is proof that it is an equation which merely expresses symbolically which operations to use as soon as defined functions are treated in place of their undefined [symbols].
The simplest case where this comes in is d(uz), where u and z are both variables, but both at the same time are functions of the same 3rd variable, e.g. of x.
If we have here obtained by means of the process of differentiation (Differenzierungsprozessi) (see the beginning of this in Book I, repeated on p.10 of this book*)
dy/dx = x⋅du/dx + u⋅dz/dx ,
then we should not forget that u and z are here both variables, dependent on x, so y is only dependent on x, because on u and z. Where with one dependent variable we had it on the symbolic side, we now have the two variables u and z on the right-hand side, both independent with respect to y but both dependent on x, and their character [as] variables dependent on x appears in their respective symbolic coefficients du/dx and dz/dx. If we deal with dependent variables on the right-hand side, then we must necessarily also deal with the differential coefficients on that side.
From the equation
dy/dx = z⋅du/dx + u⋅dz/dx
it follows:
d(uz) or dy = z⋅du + u⋅dz .
This equation only indicates, however, the operations to perform (sobald) u and z are given as defined functions.
The simplest possible case would be, for example,
u = ax, z = bx .
Then
d(uz) or dy = bx⋅adx + ax⋅bdx .
We divide both sides by dx, so that:
dy/dx = abx + bax = 2abx
and
d²y/dx² = ab + ba = 2ab .
If we take, however, the product from the veryh beginning,
y or uz = ax⋅bx = abx² ,
then
uz or y = abx², dy/dx = 2abx , d²y/dx² = 2ab .
As soon as we obtain a formula such as, for example, [w =] z⋅du/dx , it is clear that the equation, ‘what we might call’*2 a general operational equation, [is] a symbolic expression of the differential operation to be performed. If for example we take [the] expression y⋅dx/dy, where y is the ordinate and x the abscissa, then this is the general symbolic expression for the subtangent of an arbitrary curve (exactly as d(uz) = z⋅du + u⋅dz is the same for differentiation of the product of two variables which themselves depend on a third). So long, however, as we leave the expression as it is leads to nothing further, although we have the meaningful representation for dx, that it is the differential of the abscissa, and for dy, that it is the differential of the ordinate.
In order to obtain any positive result we must first take the equation of a definite curve, which gives us a definite value for y in x and therefore for dx as well, such as, for example, y² = ax, the equation of the usual parabola; and then by means of differentiation we obtain 2ydy = adx; hence dx = 2ydy/a . If we substitute this definite value for dx into the general formula for the subtangent, y⋅dx/dy, we then obtain
(y⋅2ydy/a)/dy = y⋅2ydy/ady = 2y²/a ,
and since y² = ax, [this]
= 2ax/a = 2x ,
which is the value of the subtangent of the usual parabola; that is, it is = 2 × the abscissa. If, however, we call the subtangent τ, so that the general equation runs y⋅dx/dy = τ, and y dx = τ dy. From the standpoint of the differential calculus, therefore, the question is usually (with the exception of Lagrange) posed thus: to find the real value for dy/dx.
The difficulty becomes evident if we then substitute the original form 0/0 for dy/dx etc.
dy/dx = z⋅du/dx + u⋅dz/dx
appears as
0/0 = z⋅0/0 + u⋅0/0 ,
an equation which is correct but leads nowhere (zu nichts), all the less so, since the three 0/0’s come from different differential coefficients whose different derivations are no longer visible. But consider:
1) Even in the first exposition with one independent variable, we first obtain
0/0 or dy/dx = f’(x) ; so that dy = f’(x)dx .
But since
dy/dx = 0/0, dy = 0 and dx = 0, so that 0 = 0 .
Although we again substitute for dy/dx its indefinite expression 0/0 we nonetheless commit here a positive mistake, for 0/0 is only found here as the symbolic equivalent of the real value f’(x), and as such is fixed in the expression dy/dx ,and thus in dy = f’(x)dx as well.
2) (u1 - u)/(x1 - x) becomes du/dx or 0/0 , because the variable x becomes = x1, for (u1 - u)/(x1 - x) ; we know however in general that 0/0 can have any value, and that in a specific case it has the specific value (Spezialwert) which appears as soon as a defined function of x enters for u; we are thus not only correct in substituting du/dx for 0/0 , but rather we must do it, since du/dx as well as dz/dx appear here only as symbols for the differential operations to be performed. So long as we stop with the result
dy/dx = z⋅du/dx + u⋅dz/dx ,
so that
dy = z⋅dy + u⋅dz ,
then du/dx, dz/dx, du and dz also remain indefinite values, just like 0/0 capable of any value.
3) In the usual algebra 0/0 can appear as the form for expressions which have a real value, even though 0/0 can be a symbol for any quantity. For example, given (x² - a²)/(x - a), we set x = a so that x - a = 0 and x² = a² , and therefore x² - a² = 0. We thus obtain
(x² - a²)/(x - a) = 0/0 ;
the result so far is correct; but since 0/0 may have any value it in no way proves that (x² - a²)/(x - a) has no real value.
If we resolve x² - a² into its factors, then it = (x + a) (x - a) ; so that
(x² - a²)/(x - a) = (x + a)⋅(x - a)/(x - a) = x + a ;
so if x - a = 0, them x = a, so therefore x + a = a + a = 2a.38
If we had the term P(x - a) in an ordinary algebraic equation, then if x = a, so that x - a = 0, then necessarily P(x - a) = P⋅0 = 0; just as under the same assumptions P(x² - a²) = 0. The decomposition of x² - a² into its factors (x + a) (x - a) would change none of this, for
P(x + a) (x - a) = P(x + a)⋅ 0 = 0 .
By no means, however, does it therefore follow that if the term P⋅(0/0) had been developed by setting x = a, its value must necessarily be = 0.
0/0 may have any value because 0/0 = X always leads to: 0 = X⋅0 = 0; but just because 0/0 may have any value it need not necessarily have the value 0, and if we are acquainted with its origin we are also able to discover a real value hidden behind it.
So for example P⋅(x² - a²)/(x - a) , if x = a, x - a = 0 and so as well x² = a² , x² - a² = 0; thus
P.(x² -a²)/(x - a) = P⋅0/0 .
Although we have obtained this result in a mathematically completely correct manner, it would nonetheless be mathematically false, however, to conclude without further ado that P⋅0/0 = 0, because such an assumption would imply that 0/0 may necessarily have no value other than 0, so that
P⋅0/0 = P⋅0 .
It would be more relevant to investigate whether any other result arises from resolving x² - a² into its factors (x + a) (x - a); in fact, this transforms the expression to
P⋅(x + a)⋅(x - a)/(x - a) = P⋅(x + a)⋅1,
and [when] x = a to P⋅2a or 2Pa. Therefore, as soon as we operate (rechnen) with variables,39 it is all the more not only legitimate but indeed advisable to fix firmly (festzuhalten) the origin of 0/0 by the use of the differential symbols dy/dx , dz/dx, etc., after we have previously (ursprünglich) proved that they originate as the symbolic equivalent of derived functions of the variables which have run through a definite process of differentiation. If they are thus originally (ursprünglich) the result of such a process of differentiation, then they may for that reason well become inversely (umgekehrt) symbols of a process yet to be performed on the variables, thus operational symbols (Operationssymbolen) which appear as points of departure rather than results, and this is their essential use (Dienst) in differential calculus. As such operational symbols they may even convey the contents of the equations among the different variables (in implicit functions 0 stand from the very beginning on the right-hand side [of the equation] and the dependent as well as independent variables, together with their coefficients, on the left).
Thus it is in the equation which we obtain:
d(uz)/dx or dy/dx = zdu/dx + udz/dx .
From what has been said earlier it may be observed that the dependent functions of x, z and u, here appear unchanged as z and u again; but each of them is equipped (ausgestattet) with the factor of the symbolic differential coefficient of the other.
The equation therefore only has the value of a general equation which indicates by means of symbols which operations to perform as soon as u and z are given respectively, as dependent variables, two defined functions of x.
Only when [we] have defined functions of [x] for u and z may du/dx ( = 0/0 ) and dz/dx ( = 0/0 ) and therefore dy/dx ( = 0/0 ) as well become 0, so that the value 0/0 = 0 cannot be presumed but on the contrary must have arisen from the defined functional equation itself.
Let, for example, u = x³ + ax²; then
(0/0) = du/dx = 3x² + 2ax ,
(0/0)1 = d²u/dx² = 6x + 2a ,
(0/0)2 = d³u/dx³ = 6 ,
(0/0)3 = d4u/dx4 = 0 ,
so that in this case 0/0 = 0.
The long and the short of the story is that here by means of differentiation itself we obtain the differential coefficients in their symbolic form as a result, as the value of [dy/dx in] the differential equation, namely in the equation
d(uz)/dx or dy/dx z⋅du/dx + u⋅dz/dx .
We now know, however, that u = a defined function of x, say f(x). Therefore (u1 - u)/(x1 - x) , in its differential symbol du/dx, is equal to f’(x), the first derived function of f(x). Just so z = φ(x), say, and so similarly dz/dx = φ’(x), ditto - of φ(x). The original function itself, however, provides us neither with u nor with z in any defined function of x, such as, for example
u = xm, z = sqrt{x}
It provides us u and z only as general expressions for any 2 arbitrary functions of x whose product is to be differentiated.
The equation states that, if a product, represented by uz, of any two functions of x is to be differentiated, one is first to find the real value corresponding to the symbolic differential coefficient du/dx, that is the first derived function say of f(x), and to multiply this value by φ(x) = z; then similarly to find the real value of dz/dx and multiply [it] by f(x) = u; and finally to add the two products thus obtained. The operations of differential calculus are here already assumed to be well-known.
The equation is thus only a symbolic indication of the operations to be performed, and at the same time the symbolic differential coefficients, du/dx, dz/dx here stand for symbols of differential operations still to be completed in any concrete case, which they themselves were originally derived as symbolic formulae for already completed differential operations.
As soon as they have taken on (angenommen) this character, they may themselves become the contents of differential equation, as, for example, in Taylor’s Theorem:
y1 = y + (dy/dx)⋅h + etc.
But then these are also only general, symbolic operational equations. In this case of the differentiation of uz, the interest lies in the fact that it is the simplest case in which - in distinction to the development of those cases where the independent variable x has only one dependent variable y - differential symbols due to the application of the original method itself are placed as well on the right-hand side of the equation (its developed expression), so that at the same time they enter as operational symbols and as such became the contents of the equation itself.
This role, in which they indicate operations to be performed and therefore serve as the point of departure, is their characteristic role in a differential calculus already operating (sich bewegenden) on its own ground, but it is certain (sicher) that no mathematician has taken account of this inversion, this reversal of roles, still less has it been necessary to demonstrate it using a totally elementary differential equation. It has only been mentioned as a matter of fact that, while the discoverers of the differential calculus and the major part of their followers make the differential symbol the point of departure for calculus, Lagrange in reverse makes the algebraic derivation40 of actual (wirklichen) functions of the independent variable the point of departure, and the differential symbols into merely symbolic expression of already derived functions.
If we once more return to d(uz), we obtained next as the result (Produkt) of setting x1 - x = 0, as the result of the differential operation itself:
dy/dx = z⋅du/dx + u⋅dz/dx .
Since there is a common denominator here, we thus obtain as a reduced expression
dy = z⋅du + u⋅dz .
This compares to (entspricht) the fact that in the case of only one dependent variable we obtain as the symbolic expression of the derived function of x, of f’(x) (for instance, of maxm-1, which is f’(x) if axm = f(x)), dy/dx on the left-hand side as its symbolic expression
dy/dx =f’(x)
and of which the first result is
dy = f’(x)dx
(for example, dy/dx = maxm-1; dy = maxm-1dx, which is the differential of the function y) (which last we may equally re-transform to dy/dx = maxm-1). But the case
dy = zdu + udz
it distinguished once again by reason of the fact that the differentials du, dz here lie on the right-hand side, as operational symbols, and that dy is only defined after the completion of the operations which they indicate. If
u = f(x), z = φ(x)
then we know that we obtain for du
du = f’(x)dx
and for [dz]
dz = φ’(x)dx
Therefore:
dy = φ(x)f’(x)dx + f(x)φ’(x)dx
and
dy/dx = φ(x)f’(x) + f(x)φ’(x) .
In the first case therefore first the differential coefficient
dy/dx = f’(x)
is found and then the differential
dy = f’(x)dx .
In the second case first the differential dy and then the differential coefficient dy/dx. In the first case, where the differential symbols themselves first originate from the operations performed with f(x), first the derived function, the true (wirkliche) differential coefficient, must be found, to which dy/dx stands opposite (gegenübertrete) as its symbolic expression; and only after it has been found can the differential (das Differential) dy = f’(x)dx be derived.
It is turned round (umgenkehrt) in dy = zdu + udz .
Since du, dz appear here as operational symbols and clearly indicate operations which we already know, from differential calculus, how to carry out, therefore we must first, in order to find the real value of dy/dx, in every concrete case substitute for u its value in x, and for z ditto - its value in x - in order to find
dy = φ(x)f’(x)dx + f(x)φ’(x)dx;
and then for the first time division by dx provides the real value of
dy/dx φ(x)f’(x) + f(x)φ’(x) .
What is true for du/dx , dz/dx , dy/dx ,d²y/dx² etc. is true for all complicated formulae where differential symbols themselves appear within general symbolic operational equations.
34
This excerpt is taken from notebooks which Marx entitled ‘A. I’ and ‘B (continuation of A). II’ (see pp.459, 464 of Yanovskaya 1968). It begins on the last (unnumbered by Marx) page of the notebook ‘A.I’ and ‘B (continuation of A).II’ (see pp.459, 464 of Yanovskaya, 1968. It begins on the last (unnumbered by Marx) page of the notebook ‘A. I’ and is inserted at various places in the notebook ‘B’ (Marx distinguished it with special markings). Part of the indicated draft was first published in Russian in 1933 (see Under the Banner of Marxism [Pod znamenem marksizma] No.1 as well as Marxism and Science [Marksizm I estestvoznanie], pp.34-43).
35
Marx everywhere calls ‘symbolic’ (as distinct from ‘algebraic’; see note 6) those expressions which contain the symbols specific to differential calculus, dx, dy etc. He calls ‘real’ those expressions of functions which do not contain such symbols.
36
The ‘operational formulae of differential calculus’ here means those symbolic expressions which indicate (see the text below) which operations must be performed on a defined function to obtain the real value of one or another derivative.
37
The notebook ‘A. I’ ends at this point. At the end of the page is written in Marx’s hand, ‘Sieh weiter Heft II, p.9’ (‘See further notebook II, p.9’). This indicates the notebook ‘B (continuation of A)’.
*
See p.39 of this volume.
*2
In English in original text - Trans.
38
Concerning the characteristics of this type of predefinition by continuity and the possibilities of other predefiniton satisfying these or other conditions, see note 18 and Appendix I, p.146.
39
That is, when we make the transition from the region of the usual algebra to a function (the dependent variable) for which it is necessary to predefine the ratio
(f(x1) - f(x))/(x1 - x) ,
which transforms to 0/0 at x1 = x .
40
Marx usually calls expressions not containing symbols specific to differential calculus ‘algebraic’ (see note 6) or ‘real’ (see note 2). Here and in several other passages he calls them ‘actual’ (wirkliche). Since in Russian mathematical literature the term ‘actual’ (number) carries another meaning [namely ‘real number’ - Trans], the word ‘actual’ (expression) is translated as its synonym ‘real’ [that is in Russian translation; in English ‘actual’ is not confusing - Trans].