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.

If therefore in Taylor’s theorem^{82} 1) we adopt the idea from a specific form of the binomial theorem in which it is assumed that in *(x + h) ^{m} m is a whole positive power* and thus also that the factors appear as

Summed up in general: *Taylor’s theorem* is in general only applicable to the development of functions of *x* in which *x* becomes *= x + h* or is increased from *x* to *x _{1}* if 1) the independent variable

All these conditions, however, are only another expression for the fact that this theorem *is* the binomial theorem with *whole and positive exponents*, translated into differential language.

Where these conditions are not fulfilled, where Taylor’s theorem is not applicable, that is, there enter what are called in differential calculus the *‘failures’*^{*} of this theorem.

The biggest *failure* of Taylor’s theorem, however, does not consist of these particular failures of application but rather the *general failure*, that

*y = f(x)* [and] *y _{1} = f(x + h) ,*

which are only symbolic expressions of a binomial of some sort of degree,^{84} are transformed into expressions where *f(x)* is a function of *x* which includes all degrees and thereby has *no degree* itself, so that *y _{1} = f(x + h)* equally well includes all degrees and is itself of no degree, and even more that it becomes the

This leap from *ordinary algebra*, and besides *by means of ordinary algebra*, into the *algebra of variables* is assumed as *un fait accompli*, it is not proved and is prima facie *in contradiction to all the laws* of conventional algebra, where *y = f(x), y _{1} = f(x + h)* could never have this meaning.

In other words, the starting equation

*y _{1}* or

is not only *not proved* but indeed knowingly or unknowingly assumes a substitution of *variable* for *constants*, which flies in the face of all the laws of algebra - for algebra, and thus the algebraic binomial, only admits of constants, indeed only two sorts of constants, *known* and *unknown*. The derivation of this equation from algebra therefore appears to rest on a deception.

Yet now if in fact Taylor’s theorem - whose failures in application hardly come into consideration, since as a matter of fact they are restricted to functions of *x* with which differentiation gives no result^{85} and are thus in general inaccessible to treatment by the differential calculus - has proved to be in practice the most comprehensive, most general and most successful *operational formula (Operationsformel)* of all differential calculus; then this is only the crowning of the edifice of the Newtonian school, to which he belonged, and of the Newton-Leibnitz period of development of differential calculus in general, which from the very beginning drew correct results from false premises.

The algebraic proof of Taylor’s theorem has now been given by *Lagrange*, and it in general provides the foundation *(Basis)* of *his* algebraic method of differential calculus. On the subject itself I will go into greater detail in the eventual historical part of this manuscript.^{86}

As a *lusus historiae* [an aside in the story] let it be noted here that Lagrange in no way goes back to the unknown foundation for Taylor - to the binomial theorem, the binomial theorem in the most elementary form, too, where it consists of only two quantities, *(x + a)* or here, *(x + h)*, and has a positive exponent.

Much less does he go back further and ask himself, why the binomial theorem of Newton, translated into differential form and at the same time freed of its algebraic conditions by means of a powerful blow *(Gewaltstreich)*, appears as the comprehensive, overall operational formula of the calculus he founded? The answer was simple: because from the very beginning Newton sets *x _{1} - x = dx*, so that

*Lagrange, conversely, bases himself directly on Taylor’s theorem (schliesst sich direkt an Taylor’s Theorem an)*, from a standpoint, naturally, where on the one hand the successors of the Newton-Leibnitz epoch already provide him with the corrected version of *x _{1} - x = dx*, so that as well

^{82}
This is an excerpt from the manuscript ‘Taylor’s Theorem’, which is inserted here because it contains in a more concentrated form Marx’s viewpoint on the insufficiency of the proof known to him of Taylor’s theorem, on its ‘algebraic’ origin in the binomial theorem, and on its essential difference from the latter (for more details on the unfinished ‘Taylor’s Theorem’ see p.498 [Yanovskaya, 1968]). Since the first paragraph of this extract presents difficulties in reading it in isolation from the preceding text, we note here that in this paragraph Marx summarizes the results of the previous section devoted to the critique of the proof of Taylor’s theorem in Hind’s book. In it (see Hind §74, pp.83-84; §§77-80, pp.92-96):
1) Taylor’s theorem is proved under the assumption that the expression f(x + h) may be expanded into a series of the form:
f(x + h) = Phα + Qhβ + Rhγ + ... ,
where P, Q, R, ... are functions of the variable x and the exponents α, β, γ ... are increasing positive integers.
2) The ‘cases of inapplicability’ of Taylor’s theorem are considered, with the result that for certain specific values of the variable x these conditions are not fulfilled (some of the coefficients P, Q, R ... are not defined - ‘do not have finite values’ at these points).
3) The attempt is made, following Lagrange, to show that, generally speaking, excluding, that is, certain specific values of the variable x, the conditions under which Taylor’s theorem has been proved (the exponents α, β, γ... cannot take on negative or fractional values, the functions P, Q, R... are not transformed ‘into infinity’) are fulfilled for any function f(x). After this come Marx’s remarks devoted to the insufficiency of this sort of attempt.

^{83}
The words ‘x = a, for example’ refer to the example, examined by Hind, of the expansion into a Taylor series of the expression f(x + h) where f(x) = x² + sqrt{x - a}. At x = a the expression has the intelligible value (a + h)² + sqrt{h}, but the terms of the Taylor series representing it give, according to Hind, only ‘a² + 2ah + h² + 0 + ∞ - ∞ + ∞ - etc., not at all defined’ (see Hind, p.93).

^{*}
In English in original - *Ed*.

^{84}
In the function y = f(x), where y_{1} = f(x + h) is only the symbolic expression of a binomial of a certain power, one here naturally has in mind the function y = x^{m}, where m is a positive integer.

^{85}
A literal translation of this passage would be, ‘which in the course of differentiation can give no result’ *(die auf dem Weg der Differentiation kein Resultat liefern können).*

^{86}
Literally: ‘in the possible historical part of this manuscript’ *(beim etwaigen historischen Teil dieses Manuscripts).*