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 = max ^{m-1}dx*, since here that immediately gives us the

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

*x ^{m} = f(x) = y*

to differentiate, we know immediately *(von vornherein)*

*dy = mx ^{m-1}dx*

or

*dy/dx = mx ^{m-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 expression*^{35} 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 = u _{1}z_{1}*

and

*y1 - y = u _{1}z_{1} - uz .*

Thus:

*(y1 - y)/(x _{1} - x) = u_{1}z_{1}/(x_{1} - x) - uz/(x_{1} - x)*

or

*Δy/Δx = (u _{1}z_{1} - uz)/(x_{1} - x) .*

But

*u _{1}z_{1} - uz = z_{1}(u_{1} - u) + u(z_{1} - z) ,*

since this is equivalent to

*z _{1}u_{1} - z_{1}u + uz_{1} - uz = z_{1}u_{1} - uz .*

Therefore:

*(u _{1}z_{1} - uz)/(x_{1} - x) = z_{1}⋅(u_{1} - u)/(x_{1} - x) + u⋅(z_{1} - z)/(x_{1} - x) .*

If now on both sides *x _{1} - x* becomes

*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) = x ^{m}*, a second function of

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, *mx ^{m-1}*, and is itself only the result of the operations whereby this value was derived from

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. *x ^{m}*, to differentiate, we then first look for the value of

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) *(u _{1} - u)/(x_{1} - x)* becomes

*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} = d^{4}u/dx^{4} = 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 *(u _{1} - u)/(x_{1} - x)* , in its differential symbol

*u = x ^{m}, 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 derivation^{40} 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 *x _{1} - 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 *max ^{m-1}*, which is

*dy/dx =f’(x)*

and of which the first result is

*dy = f’(x)dx*

(for example, *dy/dx = maxm-1; dy = max ^{m-1}dx*, which is the differential of the function

*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(x _{1}) - f(x))/(x_{1} - x)* ,
which transforms to