Marx's Mathematical Manuscripts 1881

# 2. From the Unfinished Manuscript ‘Taylor's Theorem’

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 theorem82 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 h = h0, h¹, h², h³, etc., that is, that h [is raised to a] whole, increasing, positive power, then 2) just as in the algebraic binomial theorem of the general form, the derived functions of x are defined and thereby finite functions in x. At this point, however, yet a third condition comes in. The derived functions of x can only be = 0, = + ∞, = - ∞, just as h[k] can only be = h-1 or hm/n (for example h1/2) when the variable x takes on particular values, x = a, for example.83

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 x1 if 1) the independent variable x retains the general, undefined form x; 2) the original function in x itself is capable of development by means of differentiation into a series of defined and thereby finite, derived functions in x, with corresponding factors of h with increasing, positive and integral exponents, so with h¹, h², h³, etc.

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] y1 = 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 y1 = f(x + h) equally well includes all degrees and is itself of no degree, and even more that it becomes the undeveloped general expression of any function of the variable x, as soon as it increases. The series development into which the ungraded f(x + h) is expanded, namely y + Ah + Bh² + ch³ + etc., therefore also includes all degrees without itself having any degree.

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), y1 = f(x + h) could never have this meaning.

In other words, the starting equation

y1 or f(x + h) = y or f(x) + Ah + Bh² + Ch³ + Dh4 + Eh5 + etc.

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 result85 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 x1 - x = dx, so that x1 = x + dx. The development of the difference is thus at once transformed into the development of a sum in the binomial (x + dx) - whence we disregard completely that it had to have been set x1 - x = Δx or = x + h). Taylor only developed this fundamental basis to its most general and comprehensive form, which only became possible once all the fundamental operations of differential calculus had been discovered; for what sense had his dy/dx, d²y/dx², etc. unless one could already develop the corresponding dy/dx, d²y/dx², etc. for all essential functions in x?

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 x1 - x = dx, so that as well y1 - y = f(x + h) - f(x), while on the other hand he produced, right in the algebraicisation of Taylor’s formula his own theory of the derived function. [In just such a manner Fichte followed Kant, Schelling Fichte, and Hegel Schelling, and neither Fichte nor Schelling nor Hegel investigated the generl foundation of Kant, of idealism in general: for otherwise they would not have been able to develop it further].

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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.
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.
In the function y = f(x), where y1 = f(x + h) is only the symbolic expression of a binomial of a certain power, one here naturally has in mind the function y = xm, where m is a positive integer.
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).
Literally: ‘in the possible historical part of this manuscript’ (beim etwaigen historischen Teil dieses Manuscripts).