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.
On a separate sheet attached to his draft sketch of the course of historical development of mathematical calculus, Marx referred to the Scholium of Lemma XI of Book One and the Lemma II of Book Two of Newton’s Principia, devoted to two fundamental concepts used by Newton throughout his mathematical analysis, the concept of ‘limit’ and ‘moment’.
In the commentary (scholium) to Lemma XI of the first book to Principia mathematica de philosophiae naturalis Newton attempts to explain the concept of ‘ultimate (limiting) ratio’ and ‘ultimate sum’ by means of a not very transparent comparison: ‘a metaphysical, not mathematical assumption,’ Marx characterised it. Indeed, Newton writes:
‘Perhaps it may be objected, that there is no ultimate ratio of evanescent quantities; because the ratio before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument it may be alleged that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, there is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place, and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of a motion may attain, but not exceed. This is the ultimate velocity. And there is a like limit in all quantities and proportions that begin and cease to be.’ (Sir Isaac Newton’s Mathematical Principles of Natural Philosophy, transl. Andrew Motte, rev. Florion Cajori, Berkeley, Univ. of Calif. Press, 1934, pp.38-39)
In present-day mathematics ‘the velocity of a body at the given moment t0’ is defined with the help of the mathematical concept of limit, and the use by science of such a definition may lead to a variety of considerations, including those of an ontological character. However, the scientific definition of the velocity of a body at a given moment by means of a certain limit of the ratio of vanishing quantities can serve neither as a demonstration of the existence of such a limit nor, a fortiori as a justification for the definition of this limit as ‘the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish,’ that is, as some sort of ratio of zeroes, the value of which is somehow compared to the speed which a body must have at the very moment when it reaches a place where its movement end. Clearly, however, from such a ‘definition’ it is impossible to extract by mathematical calculations any corresponding limit, and we are essentially in a logical circle: velocity at the moment t0 is factually described as a certain limit, the limit, itself, however, is then defined by means of the velocity àt the moment t0, the existence of which in this case now really seems to be some sort of ‘metaphysical, not mathematical, assumption’.*
Lemma II of the second book of Principia mathematica contains the following explanation of the concept of ‘moment’ (or infinitely small):
‘I understand ... the quantities I consider here as variable and indetermined, and increasing or decreasing, as it were, by a continual motion or flux; and I understand their momentary increments or decrements by the name of moments; so that the increments may be esteemed as added or affirmative moments; and the decrements as subtracted or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion, as nascent. It will be the same thing if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations and fluxions of quantities), or any finite quantities proportional to those velocities.’
It is natural that this explanation - in which Newton once again employs a ‘metaphysical, not mathematical assumption’, this time with respect to the existence of differentials (‘moments’) - should have interested Marx first of all.
But his lemma might also have attracted his attention insofar as in it Newton attempts to show the formula for the differentiation of the product of two functions without resorting to the suppression of the infinitesimals of higher order.
This (unsuccessful) attempt proceeds in the following way: Let A - (1/2)⋅a be the value of the function f(t) at the point t0, B - (1/2)⋅b be the value of the function g(t) at the same point t0, and a and b increments of the respective functions f and g on the interval [t0, t1]. (Lower we denote these Δf and Δg respectively.) Then the increment of the product f(t)⋅g(t) on the segment [t0, t1] is:
(A + (1/2)⋅a) (B + (1/2)⋅b) - (A - (1/2)⋅a) (B - (1/2)⋅b) ,
that is, Ab + Ba, which Newton also understood as the differential (‘moment’) of the derivative of the functions f and g at t0. But here Ab + Ba is not f(t0)Δg + g(t0)Δf, but
(f(t0) + (1/2)⋅Δf)Δg + (g(t0) + (1/2)⋅Δg)Δf ,
that is, different from f(t0)Δg + g(t0)Δf by the same quantity of Δf⋅Δg whose suppression Newton wanted to avoid. Identifying, although implicitly, Ab + Ba with f(t0)Δg + g(t0)Δf, however Newton in fact employed such a suppression.
As is apparent from the first drafts of the piece on the differential (see, for instance p.76), Marx at first wanted to elucidate the historical path of the development of differential calculus by the use of the example of the history of the theorem of derivative. Therefore it is not surprising that Lemma II should have drawn Marx’s attention in this connection.
Since the textbooks from which Marx made extracts do not specifically refer to Lemma XI of Book One or Lemma II of Book Two of the Principia, thee is every reason to believe that Marx selected them, having already immediately rejected Newton’s work.
Since the definition of the limit of the ratio of vanishing quantities by means of the velocity of a body at a given moment t0 contains no means for the calculation of this limit, Newton actually employs for the performance of such calculation, rather than this definition, certain hypothetical properties of limits sufficient to reduce the calculation of the limits of ratios of vanishing quantities to the calculation of the limits themselves, the numerical value of which is supposed to be completely and rigorously defined. Newton states these hypothetical properties first of all in Lemma I of the first section of Book One of Principia: ‘The method of first and last ratios of quantities, by the help of which we demonstrate the propositions that follow.’ In his notes on the history of differential calculus Marx refers to this lemma together with the scholium to Lemma XI (see pp.75 and 76).
Lemma I states: ‘Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of first time approach nearer to each other than by any given difference, become ultimately equal.’ (Newton’s Principia revised by Florion Cajori, Univ of Calif. Press, 1934, p.29)
However, in the demonstration of this limit the existence of a limit as actually reached at the end of the period of time in question is implicitly assumed. Actually, the demonstration is composed of a denial that the value of the quantities obtained ‘t the end of this time’ can be distinguished from each other.
Thus, limit is always understood by Newton in an actual sense and therefore hardly surpasses - in mathematical precision and validity - Leibnitz’s actually infinitely small differentials and their corresponding moments, which, as is well known, Newton used in practice.
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*
Consisting in that the reflection is understood as the reflected object: the contemplation in our thoughts of the anticipated goals of abstract mathematical concepts is understood as the real existence of the ideal object. - Ed.