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.
Newton’s discovery of the binomial (in his application, also of the polynomial) theorem revolutionised the whole of algebra, since it made possible for the first time a general theory of equations.
The binomial theorem, however - and this the mathematicians have definitely recognised, particularly since Lagrange - is also the primary basis (Hauptbasis) for differential calculus. Even a superficial glance shows that outside the circular functions, whose development comes from trigonometry, all differentials of monomials such as xm, ax, log x, etc. can be developed from the binomial theorem alone.74
It is indeed the fashion of textbooks (Lehrbuchsmode) nowadays to prove both that the binomial theorem can be derived from Taylor’s and MacLaurin’s theorems and the converse.75 Nonetheless nowhere - not even in Lagrange, whose theory of derived functions gave differential calculus a new foundation (Basis) - has the connection between the binomial theorem and these two theorems been established in all its original simplicity, and it is important here as everywhere, for science to strip away the veil of obscurity.
Taylor’s theorem, historically prior to that of MacLaurin’s, provides - under certain assumptions - for any function of x which increases by a positive or negative increment h,76 therefore in general for f(x±h), a series symbolic expressions indicating by what series of differential operations f(x±h) is to be developed. The subject at hand is thus the development of an arbitrary function of x, as soon as it varies.
MacLaurin on the other hand - also under certain assumptions - provides the general development of any function of x itself, also in a series of symbolic expressions which indicate how such functions, whose solution is often very difficult and complicated algebraically, can be found easily by means of differential calculus. The development of an arbitrary function of x, however, means nothing other than the development of the constant functions combined with [power of] the independent variable x,77 for the development of the variable itself should be identical to its variation, and thus to the object of Taylor’s theorem.
Both theorems are grand generalisations in which the differential symbols themselves become the contents of the equation. In place of the real successive derived functions of x only the derivatives are represented, in the form of their symbolic equivalents, which indicate just so many strategies of operations to be performed, independently of the form of the function of f(x + h). And so two formulae are obtained which with certain restrictions are applicable to all specific functions of x or x + h.
f(x + h) or y1
= y + (dy/dx)⋅h + (d²y/dx²)⋅(h²/1⋅2) + (d³y/dx³)⋅(h³/1⋅2⋅3) + (d⁴y/dx⁴)⋅(h⁴/1⋅2⋅3⋅4) + etc.
f(x) or y
= (y) + (dy/dx)⋅(x/1) + (d²y/dx²)⋅(x²/1⋅2) + (d³y/dx³)⋅(x³/1⋅2⋅3) + (d⁴y/dx⁴)⋅(x⁴/1⋅2⋅3⋅4) + etc .
The mere appearance here shows what one might call, both historically and theoretically, the arithmetic of differential calculus , that is, the development of its fundamental operations is already assumed to be well-known and available. This should not be forgotten in the following, where I assume this acquaintance.
MacLaurin’s theorem may be treated as a special case of Taylor’s theorem.
With Taylor we have
y = f(x) ,
y1 = f(x + h) = f(x) or y + (dy/dx)⋅h + (1/2)⋅(d²y/dx²)⋅h² + etc.
+ [1/1⋅2⋅3...⋅n](dny/dxn)⋅hn + etc.
If we set x = 0 in f(x + h) and on the right-hand side as well, in y or f(x) and in its symbolic derived functions of the form dy/dx, d²y/dx² etc., so that they consist simply of the development of the constant elements of x,78 then:
f(h) = (y) (dy/dx)⋅h + (d²y/dx²)⋅(h²/1⋅2) + (d³y/dx³)⋅(h³/1⋅2⋅3) + etc.
y1 = f(x + h) = f(0 + h) then becomes the same function of h which y = f(x) is of x; since h goes into f(h) just as x goes into f(x) and (y) into (dy/dx), all trace of x is wiped out.
We therefore can replace h with x on both sides and then obtain:
f(x) = (y) or f(0) + (dy/dx)⋅x + (d²y/dx²)⋅(x²/1⋅2) + etc.
+ (dny/dxn)⋅(xn/1⋅2⋅3...⋅n) + etc.
Or as others have written it,
f(x) = f(0) + f’(0)⋅x + f’’(0)⋅x²/1⋅2 + f’’’(0)⋅x³/1⋅2⋅3 + etc.
such as for example in the development of f(x) or (c + x)m:
(c + 0)m = f(0) = cm ,
m(c + 0)m-1x = mcm-1x = f’(0)x etc.
In the following, where we come to Lagrange, I will no longer consider MacLaurin’s theorem as merely a special case of Taylor’s. Let it only be noted here that it has its so-called ‘failures’* just like Taylor’s theorem. The failures all originate in the former in the irrational nature of the constant function, in the latter in that of the variable.79
It may now be asked:
Did not Newton merely give the result to the world, as he does, for example, in the most difficult cases in the Arithmetica Universalis, having already developed in complete silence Taylor’s and MacLaurin’s theorems for his private use from the binomial theorem, which he discovered? This may be answered with absolute certainty in the negative: he was not one to leave to his students the credit (Aneignung) for such a discovery. In fact he was still too absorbed in working out the differential operations themselves, operations which are already assumed to be given and well-known in Taylor and MacLaurin. Besides, Newton, as his first elementary formulae of calculus show, obviously arrived at them at first from mechanical points of departure, not those of pure analysis.
As for Taylor and MacLaurin on the other hand, they work and operate from the very beginning on the ground of differential calculus itself and thus had no reason (Anlass) to look for its simplest possible algebraic starting-point, all the less so since the quarrel between the Newtonians and Leibnitzians revolved about the defined, already completed forms of the calculus as a newly discovered, completely separate discipline of mathematics, as different from the usual algebra as Heaven is wide (von der gewöhnlichen Algebra himmelweit verschiednen).
The relationship of their respective starting equations to the binomial theorem was understood for itself, but no more than, for example, it is understood by itself in the differentiation of xy or (x/y) that these are expressions obtained by means of ordinary algebra.
The real and therefore the simplest relation of the new with the old is discovered as soon as the new gains its final form, and one may say the differential calculus gained this relation through the theorems of Taylor and MacLaurin. Therefore the thought first occurred to Lagrange to return the differential calculus to a firm algebraic foundation (auf strikt algebraische Basis). Perhaps his forerunner in this was John Landen, an English mathematician from the middle of the 18th century, in his Residual Analysis. Indeed, I must look for this book in the [British] Museum before I can make a judgement on it.
Lagrange proceeds from the algebraic basis (Begründung) of Taylor’s theorem, and thus from the most general formula of differential calculus.
It is only too noticeable with respect to Taylor’s beginning equation:
y1 or f(x + h) = y or f(x) + Ah + Bh² + Ch³ + etc.
1) This series is in no way proved; f(x + h) is no binomial of a defined degree; f(x + h) is much more the undefined general expression of any function [of the variable] x which increases by a positive or negative increment h; f(x + h) therefore includes functions of x of any defined degree but at the same time excludes any defined degree to the series expansion itself. Taylor himself therefore puts ‘+ etc.’ on the end of the series. However, that the series expansion which is valid for defined functions of x containing an increment - whether they are capable of representation now in a finite equation80 or an infinite series - is no longer applicable to the undefined general f(x) and therefore equally well to the undefined general f(x1) or f(x + h), must first be proved.
2) The equation is translated into the language of differentials by virtue of the fact that it is twice differentiated, that is, y1 once with respect to h as variable and x constant, but then again with respect to x as variable and h constant. In this manner two equations are produced whose first sides are identical while their second sides are different in form. In order, however, to be able to equate the undefined coefficients (which are all functions of x) of these two sides, it is also necessary to assume both that the individual coefficients A, B, etc., are undefined, to be sure, but finite quantities, and that their accompanying factors h increase in whole and positive powers.81 If it is assumed - which is not the case - that Taylor had proved everything for f(x + h) as long as the x in f(x) remains general, then for that very reason it would not be valid at all as soon as the functions of x took on definite, particular values. This could be on the contrary irreconcilable with the treatment, by means of its series,
y1 = y + (dy/dx)⋅h (d²y/dx²)⋅h² + etc.
In one word, the conditions or assumptions which are involved in Taylor’s unproven beginning equation are naturally found also in the theorem derived from it:
y1 = y + (dy/dx)⋅h + (d²y/dx²)⋅h² + etc.
It is therefore inapplicable to certain functions of x which contradict any of the assumptions. Therefore the so-called failures of the theorem.
Lagrange provided an algebraic foundation for the beginning equation (begründet die Ausgangsgleichung algebraisch) and at the same time showed by means of the development itself which particular cases, due to their general character, that is, contradicting the general, undefined character of the function of x, are excluded.
H) 1) Lagrange’s great service is not only to have provided a foundation in pure algebraic analysis for the Taylor theorem and differential calculus in general, but also and in particular to have introduced the concept of the derived function, which all of his successors have in fact used, more or less, although without mentioning it. But he was not satisfied with that. He provides the purely algebraic development of all possible functions of (x + h) with increasing whole positive powers of h and then attributes to it the given name (Taufname) of the differential calculus. All the conveniences and condensations (Taylor’s theorem, etc.) which differentials calculus affords itself are thereby forfeited, and very often replaced by algebraic operations of much more far-reaching and complicated nature.
2) As far as pure analysis is concerned Lagrange in fact becomes free from all of what to him appears to be metaphysical transcendence in Newton’s fluxions, Leibnitz’s infinitesimals of different order, the limit value theorem of vanishing quantities, the replacement of 0/0 ( = dy/dx ) as a symbol for the differential coefficient, etc. Still, this does not prevent him from constantly needing one or another of these ‘metaphysical’ representations himself in the application of his theories and curves etc.
In the manuscripts devoted to the history of differential calculus there are two passages, located almost immediately adjacent to one another, at which Marx proposed to insert: 1) an investigation of the theorems of Taylor and MacLaurin and 2) a discussion of Lagrange’s theory of analytic functions (see p.97). Marx did not succeed in accomplishing his intentions, although he had in his possession a great deal of material on these subjects which he had collected from his sources and which served as the foundation from which he arrived at the point of view on the essence of differential calculus which he presented in the works conveyed to Engels. This material is comprised primarily of outlines but also includes manuscripts containing Marx’s summarizing or critical comments. The most important of these comments are contained in the manuscripts: 1) ‘Taylor’s Theorem, MacLaurin’s Theorem, and Lagrange’s Theory of Derived Functions’ (for more details see p.441 [Yanovskaya, 1968]) and 2) ‘Taylor’s Theorem’ (unfinished), extracts from which are reproduced here, in order to amplify somewhat Marx’s intentions mentioned above. For extracts from other outlines on the same subjects see pp.281, 412 [Yanovskaya, 1968].
74 In the handbook on differential calculus at Marx’s disposal the derivatives of all elementary functions, except for the trigonometric ones, were actually calculated by means of the binomial theorem. Marx noted this himself in his manuscript, ‘Theorems of Taylor and MacLaurin, First Systematisation of Material’ (see pp.419-420 [Yanovskaya, 1968]). Subsequently Marx formulated for this class of function a different means of differentiation which he called the ‘algebraic’ (see the manuscript ‘On the Concept of the Derived Function’). Therefore it is clear that the present manuscript chronologically precedes ‘On the Concept of the Derived Function’ and ‘On the Differential’.
75 Thus, in Hind’s textbook (Hind, pp.84-85), after the example containing the derivation of the binomial theorem by means of the expansion of (x + h)m into the Taylor series there is introduced the derivation of the theorems of Taylor and MacLaurin from the binomial theorem.
76 Here (see also p.514 [Yanovskaya, 1968]) Marx says straight out, that by ‘increment’ of the value of the variable x he has in mind any change of this value, whether it be a positive or negative increment h.
77 Because, according to Marx, a function in x is a given expression, it represents a combination of symbols which is considered with respect to the appearance in it of the variable x. In the given case we have before us the terms of the MacLaurin series, that is the product (‘combination’) of the two expressions, 1) xk (k = 0, 1, 2, 3 …) and 2) its corresponding ‘constant function’ f[k](0)/h(?).
78 Marx calls expressions not containing the variable x ‘constant functions’ of x. (y), (dy/dx), (d²y/dx²) and so on are expressions for f(x) and its successive derivatives in which all appearances of the variable x have been replaced by a constant - zero. The result of this substitution in y and in its corresponding derivative dyk/dxk is designated in the manuscripts as (y) and, correspondingly, (dyk/dxk). This designation, which Marx borrowed from Boucharlat (see Boucharlat, p.40), has been preserved in this edition.
* In English in quotes in the original - Trans.
79 Marx does not explain what exactly he means here by ‘the irrational nature of the constant (or variable) function’. Apparently it deals with the fact that in both cases the cause of the origin of ‘exceptions’ is the presence in the expansion of terms having no rational mathematical meaning: in the first case without any continuity (such as, for example , a ‘fraction’ of the form 0/0), and in the second without defined values of the variable x (such as, for example, c/(x - a) at x = a). The ‘irrationality’ of such an expression does not imply that it necessarily contains a radical sign (compare ‘algebraic irrationality’), but is used as the opposite of intelligibility (rationality; compare the ‘rational differential calculus of Euler and d’Alembert’ with the opposite ‘mysticism of Newton and Leibnitz’!). Marx gives a short general characterisation of cases of inapplicability at the very end of the manuscript, ‘Theorems of Taylor and MacLaurin, First Systematisation of Material’ (see pp.440 [Yanovskaya, 1968]).
80 By ‘representation in a finite equation’ here is obviously meant a representation of the form f(x + h) = P0 + P1h + P2h² + … + Pnhn, where n is a positive integer, and P1(i = 0, 1, 2, ... n) are functions of x.
81 For a more detailed exposition of the proof of Taylor’s theorem contained in the sources used by Marx, an exposition necessary in order to understand the critique to which Marx subjected it in the following lines, see Appendix VI, p.182.