Marx's Mathematical Manuscripts 1881

Appendix I. Concerning the Concept of ‘Limit’ in the Sources consulted by Marx

Written: August, 1881;
Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Russian translation, in Pod znamenem marksizma, 1933.

In order to give the reader accustomed to the contemporary use in mathematics of the term ‘limit’ a correct understanding of Marx’s critical remarks concerning this concept and of Marx’s interpretation of it, we give first of all the definition of ‘limit’ (and clarifying examples) and the ways of using the word ‘limit’ contained in the courses of Hind and Boucharlat which Marx possessed and studied critically.

Hind’s course-book follows d’Alembert, which is to say that the derivative was defined in it by means of the concept of limit. The introductory chapter of the book was therefore entitled ‘The method of limits’. However, neither in this chapter nor in the rest of the textbook was there a definition of ‘limit’. There were only definitions of the ‘limits’ of a variable in the restricted sense of the exact upper or lower bounds to the multiplicity of its value. (This multiplicity might include, in particular, an ‘infinitely large’ value of the variable, designated by the symbol ∞. But there were no precisely defined correct operations with this symbol: there was no concept of absolute value, no +∞ and -∞; it was considered simply self-evident, that for any α ≥ 0, ∞ + α = ∞, that for any finite a (that is, distinct from 0, as well as from ∞) a⋅∞ = ∞ and a/∞ = 0. )

This concept of the limit of a function - a concept which of course can only be surmised from the examples - was introduced in the introductory chapter, implicitly, by means, as might be anticipated, of identifying this limit (at the point coinciding with the exact upper or lower bounds of the given multiplicity of the values of the argument) with one of the ‘limits’ (with the exact upper or with the exact lower bound) of the corresponding multiplicity of values of the function. Since only monotonic or piecewise monotonic functions are examined in this book, such a ‘limit’ appears in practice to be with the (one-sided) limit in the more usual sense of the word, in which Hind actually uses the concept of limit in all the remaining parts of the book. It turned out, however, that the introduction of this concept, which was supposed to ‘improve’ the method of infinitely small quantities, did not consciously attain that goal and was generally unwarranted.

Actually, Hind might have replaced the evaluation of the one-sided limit of a piecewise monotonic function f(x), defined on the interval (a, b) by the solution of the following two problems as x moves to + a:

1. To find a certain number α such that for a < x < α the functions is monotonic (in the broad sense, i.e., non-decreasing or non-increasing; for demonstration we will assume the function is here monotonically non-decreasing);

2. To evaluate the point at the (by our assumption lower) boundary of the possible values of the function on the interval (a, α), that is, for a < x < α. Clearly, this will be the desired lim x→+ a f(x).

But Hind did not proceed in this manner. Following Newton (see the appendix ‘On the lemmas of Newton cited by Marx’) he considered the limit simply the ‘last’ value of the function of the ‘last’ value of the independent variable. In other words he looked at lim x→+a f(x) as the point of the lower boundary of the values of the function not on the interval a < x < α but on the segment a ≤ x ≤ α. He assumed the ‘last’ value f(a) to be already defined; but in that case all of the above procedure loses meaning, since α may take the value a, and to find the lower boundary of all possible values of the function, consisting now of only the one f(a), now becomes that same f(a).

This was just what Marx wanted to say, apparently, when he noted, obviously having in mind Hind’s determination, that it is meaningless to treat 3x² as the limit value of the function 3x² has h goes to zero, later terming such treatment a ‘well-worn tautology’ (see pp.124-6 and notes 90-92); where he calls generally ‘childish’ and ‘the origin of the first mystical and mystifying method of calculus’ (see p.126) the actual approach to the limit, the assumption, that the limit value of the function is formed as its ‘last’ value at the ‘last’ value of the argument.

This circumstance, that the actual approach to the limit by no means resolves the difficulties surrounding infinitely small quantities, becomes particularly evident in the case when the ‘last’ value of the independent variable is ‘infinity’. So, in particular, if we consider the sequence {an}, then the limit must be that member of the series for which n = ∞; so we regard a limit as the end (the last term) of an infinite (that is, without an end) series of terms. It is hardly surprising that this concept of the ‘actual limit’ should be no clearer than the concept of ‘infinitely small quantities’ which Marx called ‘mystical’.

As is well known, the definition of the limit of a function, not requiring the carrying-out of an infinite number of steps and permitting an exact formulation in terms of only finite variables and parameters, gained currency is mathematics only after the time of Cauchy, that is, in the 70s of the last century. But even at this time the authors of many widely-distributed textbooks did not clearly understand that the limit was not be interpreted actually; that even in cases where the function is continuous at the point a, that is, the limit of the function f(x) as x→a is equal to f(a), nevertheless it must be shown equal to f(a) on the condition that, no matter how closely x approaches a, it never reaches it.

With regard to Marx’s mathematical manuscripts it is essential for us to note, that if the value f(a) is undefined but the limit f(x) exists as x→a (corresponding to x over the interval (a - k, a + k)) then we may simply predefine the function of f(x) at the point a, f(a), as that limit, by definition. Such predefinition of the value of the functions is also a predefinition of continuity. The limit of the function f(x) as x→a would in this case be the value of the already well-defined function with x = a. This however does not mean that one may treat the value f(a) as the determination of the known single-valued function f(x), but on the contrary only as a quantity at the end of an infinite progression no matter how closely x approaches a. Indeed, Marx himself obviously had such a predefinition of ‘continuity’ in mind when he called the limit of the expression Δy/Δx as Δx→0, the ‘absolute minimal expression’ of he ratio (see, for example, p.125); by this he graphically had in mind the limit of this ratio as Δx→0 under the condition that there exists a certain number α, such that for 0 < Δx < α as Δx decreases so does the ratio Δy/Δx. By means of this definition of a function Lacroix works out the example he gives (see below p.153). But even so far in the construction of mathematical analysis as Lacroix had gone beyond the metaphysical ‘principle of continuity’ of Leibnitz, which he regarded as a self-evident axiom, nonetheless he did not consider any other definition of function generally possible. Regarding the fact that Marx quite obviously allowed other means of definition of the ratio Δy/Δx as Δy = Δx = 0, see p.18 and note 18.

We now give some of Hind’s own words which may be necessary in reading Marx’s manuscripts and from which follow the conclusions set out above.

In his introductory chapter ‘On the method of limits’ Hind begins with definition number one, to wit:

‘By the limits of a quantity allowed to vary in value we intend those values, between which are contained all those values which it may have throughout all its changes; beyond which it may not extend and distinct from which may be made the quantity; - provided that they can be expressed in finite terms’ (that is, without the use of the symbols 0 and/or ∞ - S.A. Yanovskaya. See Hind, p.1, our italics.)

With the definition there follows a series of examples, in which, however, not once is brought into clear view nor once is demonstrated that the ‘limit’ spoken of by the author actually fulfils the requirements formulated in Definition One. The first of these examples is the following:

‘The quantity ax, wherein x admits of all possible values from zero or 0, to infinity, or ∞, becomes 0 in the former case and ∞ in the latter; and consequently the limits of the algebraical expression ax are 0 and ∞: the first is the inferior, the second the superior limit.’ (Sic. It is here obviously assumed that a > 0.)

Already the first example plunges the student into confusion. How can the quantity ax be made to differ from the value ∞ by finite quantities, ‘a magnitude from which it may be made to differ by quantities less than any that can be expressed in finite terms’? Indeed, following Hind, when x assumes a finite value the difference ∞ - ax is equal to infinity, but when x = ∞, then ax = ∞, and the difference ∞ - ∞ is undefined.

In the second example (it is necessary to consider, naturally, the values In these conditions of x and a respectively) the lower and upper limits of the expression ax + b are found, appropriately enough, at b and infinity.

In the third example the lower limit of the fraction (ax + b)/(bx + a), that is, b/a is found by simple substitution of 0 in the place of x in the expression, and the upper limit, a/b, by the substitution of ∞ in place of x in the equivalent fraction (a + (b/x))/(b + (a/x)). An explanation of under what conditions the values given to a and b respectively appeared actually in the lower and upper limits does not accompany the example. There is not even a hint of the question of whether if the values are tested they will satisfy the adduced definition of ‘limits’ (to check, for example, that we are looking at monotonic functions). The reader is thus pre-‘prepared’ to find a limit to a function through the direct substitution into its expression (or into its re-arranged expression in those cases where the immediately given continuous expression is devoid of any meaning) of the limit value of the independent variable.

The fourth and the sixth examples, exactly those examples which typify point two of the introductory chapter - in which proceeds the gradual ‘transition’ from the concept of inferior and superior limits of the function tot he more conventional concept of limit and in which is revealed the actual character of limit according to Hind - we reproduce here in full. From them it will become sufficiently clear what a jumbled character is attributed to any general account of the concept of limit by this author:

‘Example 4: The sum of the geometric series

a + a/x + a/x² + etc. ,

(a(1/xn - 1))/(1/x - 1) = (ax(1 - 1/xn))/(x - 1) ;

now, if n = 0, the inferior limit is manifestly = 0; but if n = ∞, 1/xn becomes 0, and therefore the superior limit is ax/(x - 1) ; which is usally called the sum of the series continued ad infinitum.

‘Example 6. If a regular polygon be inscribed in a circle, and the number of its sides be continually doubled, it is evident that its perimeter approaches more and more nearly to equality with the periphery of the circle, and that at length their difference must become less than any quantity that can be assigned; hence therefore, the circumference of the circle is the limit of the perimeters of the polygons.’ (pp2-3)

Here one no longer speaks of one of the ‘limits’ of the sequence nor any more about the superior of the limits, as would naturally follow from Definition One, but simply of the limit in the usual sense.

‘2. To prove that the limits of the ratios subsisting between the sine and tangent of a circular arc, and the arc itself, are ratios of equality.

‘Let p and p’ represent the perimeters of two regular polygons of n sides, the former inscribed in, the latter circumscribed about, a circle whose radius is 1, and circumference = 6.28318 etc. = 2π; then (trig.)

p = 2n sin π/n, and p’ = 2n tan π/n ;


p/p’ = (2n sin π/n)/(2n tan π/n) = cos π/n ,

and if the value of n be supposed to be indefinitely increased, the value of cos π/n is 1, and therefore p = p’; now, the periphery of the circle evidently lies between p and p’, and therefore in this case is equal to either of them; hence on this supposition, an nth part of the perimeter of the polygon is equal to an nth part of the periphery of the circle: that is,

2 sin π/n = 2π/n = 2 tan π/n, or sin π/n = π/n = tan π/n ,

or the sine and the tangent of a circular arc in their ultimate or limiting state, are in a ratio of equality with the arc itself.’ (p.3)

The word ‘limit’ or ‘limits’ occurs here only in the verbal formulation of the theorem, but recalling that formulation we see that one surmises that the requirement is to show the equality of (sin x)/x and (tan x)/x as x goes to 0. However, Hind’s proof can hardly be considered satisfactory by the standards of his time. Indeed, from the above account it is evident that the author desires to show that

sin π/n = π/n = tan π/n as n = ∞ (1)

But even here, in order to have cos(π/n) = 1 when n = ∞ he already assumes that π/n = tan 0 = 0. That is, in order to prove equation (1) - from which, of course, it by no means follows by itself the theorem on the limit of the ratio (sin x)/x as x→0 - the assumptions immediately preceding the introduction by the author of this equation are missing completely.

It remains equally difficult to explain how all this confusing account could possibly demonstrate the superiority of this methods of limits, literally interpreted, over the method of infinitely small quantities, in this case simply the identification of an infinity small segment of the perimeter of the circle with its chord.

In Boucharlat’s textbook as well (see p.vii) the method of limits is treated as an improvement on the method of infinitely small quantities: ‘repairing that which may be imperfect in this last’. There is, however, no attempt in Boucharlat’s course to define what is meant by ‘tends to (such-and-such) a limit’ (or how to make certain that such-and-such a quantity actually tends towards such-and-such a limit). In it the concept of limit, as well as of ‘actual’, appears for the first time in evaluating the derivative of the function y = x². We reproduce here in full that passage which elicited critical remarks from Marx in his manuscript ‘On the ambiguity of the terms “limit” and “limiting value”.’

‘By attending to the second [right-hand] side of equation (2)

(y’ - y)/h = 3x² + 3xh + h² , (2)

we see that this ratio is diminished the more h is diminished, and that when h becomes 0 this ratio is reduced to 3x². This term 3x² is therefore the limit of the ratio (y’ - y)/h, being the term to which it tends as we diminish h.

‘Since, on the hypothesis of h = 0, the increment of y becomes also 0, (y’ - y)/h is reduced to 0/0, and consequently the equation (2) becomes

0/0 = 3x² (3)

‘This equation involves in it nothing absurd, for from algebra we know that 0/0 may represent every sort of quantity; besides which it will be easily seen, that since dividing the two terms of a fraction by the same number the fraction is not altered in value, it follows that the smallness of the terms of a fraction does not at all affect its value, and that, consequently, it may not remain the same when its terms are diminished to the last degree, that is to say, when they become each of them 0.’ (pp.2-3)

For a correct understanding of the above-mentioned manuscript of Marx it is essential to note that in Boucharlat’s account the transition from the equation of the form Δy/Δx = Φ(x1, x) (where y = f(x)) to an equation of the form dy/dx = f’(x) is presented as divided into those parts to the left and to the right in the first equation above: from Δy/Δx to dy/dx and from Φ(x1, x) to f’(x). And the limit of the ratio Δy/Δx - corresponding to the (y1 - y)/h of equation (2) - is evidently considered equivalent to the expression 0/0, denoted dy/dx. So, in his determination of the differential of x, having deduced the equation (y1 - y)/h = 1, Boucharlat concludes: ‘Since the quantity h does not enter into the second side of this equation, we see that to pass to the limit it is sufficient to change (y1 - y)/h into dy/dx which gives dy/dx = 1, and therefore dy = dx.’ (p.6)

The case where the limit appears equal to zero Boucharlat treats as equivalent to the nonexistence of a limit. So, taking the derivative of y = b and obtaining the equation dy/dx = 0, he concludes, ‘so there is neither limit nor differential’ (p.6).

Boucharlat obtains the limit of the ratio (sin x)/x as x→0 in essentially the same manner as Hind, although in a more intelligible form. He proves at first the theorem given as an example in his textbook, that ‘the arc is greater than the sine, and less than the tangent’. (p.24) However, he makes no mention of the fact that immediately follows, viz:

(sin x)/(tan x) < (sin x)/x < (sin x)/sin(x) ( 0 < x < /2) ,

that is, that the ratio (sin x)/x lies between cos x and 1. All this aside, following Hind, Boucharlat writes:

‘It follows from the above, that the limit of the ratio of the sine to the arc is unity; for since, when the arc h … becomes nothing, the sine coincides with the tangent; much more does the sine coincide with the arc, which lies between the tangent and the sine; and, consequently, we have, in the case of the limit, (sin h)/(arc h) or rather (sin h)/h = 1.’ (p.29)

The condition that for h = 0 the ratio (sin h)/h is ‘transformed’ into 0/0, that is, in general, is undefined, and the conclusion drawn on no more ground than ‘the sine coincides with the arc’ when this last is changed into zero, all these embarrass Boucharlat no more than they embarrass Hind.

We have dwelt long enough, obviously, on the treatment of the concept of limit in the textbooks fo Hind and Boucharlat in order to clarify those passages in the manuscript ‘On the ambiguity of the terms “limit” and “limiting value”’ in which Marx criticised these authors’ actual transition to the limit, (concerning which see notes 90-92).

In order to understand other passages of the manuscripts, and in particular Marx’s characteristic ratio treatment of the limit, closer to the contemporary one, it is advisable to introduce certain opinions regarding the concept of limit in other sources with which Marx familiarised himself, first of all the 3-volume Traité of Lacroix on the differential and integral calculus, 1810.

Following Leibnitz, Lacroix considered all sorts of functions obeying the requirements of the law of continuity, but considered the passage to the limit to be expression of this law, ‘c’est-à-dire de la loi qui s’observe dans la description des lignes par le mouvement, et d’après laquelle les points consécutifs d’une même ligne se succedent sans aucun intervalle.’ (p.xxv) (‘that is, the law which is observed of lines when described by [their] movement, and according to which there is not the slightest interval between successive points of the same line’). For any such change in the quantity is impossible to understand without considering its two different values, between which the interval is being considered, since the law of continuity must be expressed in term of it, that ‘plus il est petit, plus on se rapproche de la loi dont il s’agit, à laquelle la limite seule convient parfaitement’, (ibid: ‘the smaller it becomes the more closely it approaches the law which it obeys, to which only the limit fits with complete agreement’). Lacroix also explains that this role of continuity in mathematical analysis seemed to him appropriate in order to ‘employer la méthode des limites’ (p.xxiv) for the construction of a systematic course-book of mathematical analysis.

The concepts ‘infinite’ and ‘infinitely small’ Lacroix considers determined only in a negative sense, that is, as ‘l’exclusion de tout limite, soit en grandeur, soit en petitesse, ce qui n’offre qu’une suite de négations, et ne sourait jamais constituter une notion positive’ (p.19 ‘the exclusion of any limit wheter of greatness or of smallness, this only offers a series of negations and never rises to constitute a positive notion’). And in a footnote on the same page he adds ‘l’infini est necessairrement ce dont on affirme que les limites ne peuvent être atteintes par quelque grandeur conçevable que ce soit,’ (‘ the infinite is necessarily that of which one believes its limits cannot be surpaseed by any conceivable quantity no matter how large’). In other words, Lacroix does not accept any actual infinity: neither an actual infinitely large quantity nor an actual infinitely small one.

Lacroix introduces the concept of limit in the following manner:

‘Let there be given a simple function ax/(x + a) in which we suppose x to be augmented positively without end. In dividing the numerator and divisor by x the result

a/(1 + a/x) ,

clearly shows that the function will always remains less than a, but will approach that value without a halt, since the part a/x in the denominator diminished more and more and can be reduced to any degree of smallness which one would want. The difference between the given fraction and the value a is expressed

a - ax(x + a) = a²/(x + a) ,

and becomes therefore smaller and smaller as x is larger, and could be made less than any given quantity, however small; it follows that the given fraction can approach a as closely as one would want: a is therefore the limit of the function ax/(x + a) with respect to the indefinite increase of x.

‘The terms which I now am stating comprise the true value [which it is necessary to attribute to] the word limit in order to understand all of what it implies.’ (pp.13-14)

Already in Lacroix there is no longer any assumption of a monotonic or piecewise monotonic function, and his limit is not, in general, a one-sided limit: the variable may approach its limiting value in any manner whatsoever. In place of the concept of absolute value Lacroix employs, although not consistently, the expression ‘value without sign’, the meaning of which, however, remains unspecified. He emphasised that the function may not only attain its limiting value but in general may even pass beyond, to oscillate in its vicinity. But Lacroix still did not formulate in clear terms the restriction on the independent variable that in its approach to its limiting value α, related to the passage to the limit, it is assumed that it does not attain α, that is, that the limit is not to be understood actually. As long as the function with which he is concerned is continuous, that is, its limits coincide with the value of the function at the limiting value of the independent variable, he expresses himself as would a man who believed that the approach of the independent variable to its limiting value must in the passage to the limit be completed by reaching that value.

It must also be noted that Lacroix uses the same one word ‘limit’ for the designation of the limit - an end which as we have seen was conceived by him in a much more general, more precise way, and closer to the contemporary sense that anything in the concepts of the textbooks of Boucharlat and Hind which Marx criticised - as he uses in several instances for the designation of the limit value as well.

These lines on the concept of limit in the long treatise of Lacroix - which, as we know, Marx considered his most reliable source of information on the fundamental concepts of mathematical analysis, such as function, limit etc. - are obviously sufficient to clarify what Marx had in mind when he noted briefly regarding the concept of limit in Lacroix’s treatment, that ‘this category, brought into general use in [mathematical] analysis largely by Lacroix’s example, acquires great significance as a replacement for the category “minimal expression”’ (p.68). It is clear, first of all, that Marx actually understood what he was doing when he introduced, in dealing with the ambiguity of the term ‘limit’, the concept of the ‘absolutely minimal expression’, in the same sense as that which we recognise today in the concept of limit. Marx foresaw, it is also clear, that with the concept of limit as understood by Lacroix we are forced, after completely replacing, obviously, the less satisfactory concept of limit, to perform the unnecessary introduction of the special - new - concept of the ‘absolutely minimal expression’; in other words, we are faced with the necessity of replacing the latter.

It is probably appropriate, in connection with this same extract from the manuscripts of Marx which we are discussing at the moment, but also with regard to a variety of other passages of the manuscripts, to introduce the words of Lagrange with respect to the concept of limit from the introduction to his Theory of Analytic Functions (Oeuvres Lagrange, Vol IX, Paris, 1881).

Speaking about the attempts by Euler and d’Alembert to regard infinitely small differences as absolutely zero, with only their ratios entering into calculus, and to see these as the limits of the ratios of finite or indefinitely small differences, Lagrange wrote (p.16):

‘Mais il faut convenir que cette idée, quoique just en elle-même, n’est pas assez claire pour servir de principe à une science dont la certitude doit être fondée sur l’evidence, et surtout pour être presentée aux commençants.’ (‘But it is necessary to admit that this idea, however correct in itself, is not at all clear enough to serve as the principle of a science whose certitude must be founded solely on evidence and must above all be presentable to beginners.’)

Later (p.18) he remarks, in connection with the Newtonian method of the remaining ratios of disappearing quantities, that

‘cette méthode a, comme celle des limites dont nous avons parlé plus haut, et qui n’en est propement que la traduction algébraique, le grand inconvénient de considérer les quantités dans l’état où elles cessent, pour ainsi dire, d’être quantité, car, quoiqu’on conçoive toujours bien le rapport de deux quantités, tant qu’elles demeurent finies, ce rapport n’offre plus à l’ésprit une idée claire et precise aussitôt que ses termes deviennent l’un et l’autre nuls à la fois.’ (‘This method has, like that of limits of which we spoke earlier and of which it is only the algebraic translation, the great inconvenience of having to consider quantities in the state in which they, so to speak, cease to be quantities; since however well one understands the ratio of two quantities so long as they remain finite, such a ratio no longer presents a clear and precise idea to the understanding unless both of its terms become zero simultaneously.’)

Lagrange then turned to the attempts of ‘the clever English geometrician’ [John] Landen to deal with these difficulties, attempts which he valued highly, although he considered Landen’s method too awkward. (See Appendix IV, ‘John Landen’s Residual Analysis’, pp.165-173)

‘Of himself, Lagrange wrote that already in 1772 he maintained ‘the theory of the development of functions into a series containing the true principles of differential calculus separate from all consideration of infinitely small quantities or of limits’. (p.19)

Thus it is clear that Lagrange considered the method of limits no more perfect than the method of infinitely small quantities and that this was related to his understanding that the limit of which one speaks in analysis is understood actually as the ‘last’ value of the function for the ‘last’ (‘disappearing’) value of the independent variable.