A Mathematical letter from Engels to Marx
The following quotation is part of translation of a letter the original of which is printed in
Karl Marx/Friedrich Engels, BRIEFWECHSEL; IV. BAND: 1868-1883; Dietz Verlag, Berlin 1950.
This book is published in four volumes, and it is the fourth that is dated as published in 1950. Any citation of the whole work should say "1949-1950".
An epigraph opposite the title page reads:
"Diese Ausgabe ist ein unveranderter Nachdruck der im Jahre 1939 vom Marx-Engels-Lenin-Institut Moskau besorgten Ausgabe"
The Library of Congress number is HX 276.M39b.v4.
The letter of 18 August 1881 I quote from begins on page 610. There is another letter, beginning on page 620 and dated 21 November 1882, also concerned with the interpretation of dy/dx, but referring to an enclosure not reprinted here, by a certain Moore, to whom Engels seems to have written about it.
Translator's note: My German is not strong, even though I passed a German exam in graduate school. I call attention below to places where I knew I was out of my depth. Any man who is his own translator has a fool for an editor. My translation dates from 1996, when I first prepared this note, but part of it was taken from an article I read somewhere but can no longer remember. Unfortunately my source didn’t reproduce all that I wish to use, and I had only some notes. It is not my purpose to pass off the text as a translator’s contribution to scholarship. Furthermore, I have now discovered that a 1948 paper by Dirk Struik appeared in Science and Society 12(1), p.181-196, and is found in the anthology Ethnomathematics, where most of what I had earlier translated myself (or found in that unnamed “somewhere”) is given in Struik’s own translation. I make note below of the places where our translations differ in any important way, especially in places where I myself hadn’t had a clue and Struik, whose German must be better than mine, had.
Author’s note: My major purpose in reproducing the letter fragments here differs somewhat from that of Struik. Struik, a well-known Marxist, is trying to bring Marx into the history of mathematics, where he has no place whatever. My purpose here is to call attention to the portion of the Engels letter italicized by me below, a few sentences that Struik omitted in his own paper.
That paragraph, which is mathematically illiterate even by the standards of D’Alembert and Lagrange, to whom Marx and Engels made frequent reference, and even by the standards of Newton and Leibniz, is put forward not merely ignorantly but in the case of Engels arrogantly, accompanied by a comment on the “thick skulls” of “these one-sided gentlemen” (i.e. the professional mathematicians of their time), gentlemen who certainly could never, according to Engels, have scooped Karl Marx in announcing the solution of the problem of infinitesimals to the scientific world.
Thus the attitude of a universal philosopher. Marx didn’t believe he was unqualified to settle the serious philosophical question of his time concerning the foundations of analysis, and his friend Engels agreed, going further in his own texts than Marx apparently did (Struik’s paper has more to say about Marx’s own ideas than appears below), with his enthusiastic announcement that “0/0” answered the question of the meaning of dy/dx. It is not plain to me exactly what Marx himself had been saying, though Engels’s sycophantic approval of some papers he had just received from the master suggest that Engels, at least, regarded his own nonsense to be consistent with it. Marx, in discussing differentiation, stopped with Lagrange and (naturally) Hegel, apparently unaware of the work of Cauchy, which a serious student of the matter in 1881, the date of this correspondence, surely would have known.
To Marx, all philosophy was his domain, and he was as willing to cite a 17th Century philosopher, or Aristotle, as one of his own time. Hegel’s comments on mathematics, whatever they were, were recent enough to be taken by him as modern; and armed with dialectical materialism Marx could trump the entire mathematical profession of his time. To a true believer, Engels’s “0/0” must be as embarrassing as Hobbes’s notion that he had squared the circle, trisected the angle, and found any given numbers of mean proportionals between two lines. All of them. Really, one should leave something to the experts, or at least find out what all the experts have been saying before announcing so revolutionary an accomplishment. (One may somewhat excuse Mr. Hobbes here by noting that he made all these statements when deep into his nineties, though the part about squaring the circle he had claimed much earlier.)
One cannot say that the problem of the foundations of analysis has been settled for all time, but it is now well over a hundred years since the axiomatic basis of the real number system, as rooted in the very intuitive Peano axioms for the counting numbers and in the Zermelo-Frankel axioms for sets, has taken root in the mathematical world. I doubt that any other system will really replace it in the sense that it will show that we have been unclear in our formulation of the theory. Between Leibniz and Dedekind there was a period of centuries of controversy, and of more or less confusion, concerning the very meaning of “derivative”, and later the allied notion of “limit”, and the world of mathematics knew this. Since the “arithmetization of analysis” in the late 19th Century there has been no corresponding controversy or doubt, and the more recent developments in logic, topology and geometry have only confirmed the correctness (however “correctness” might be defined; “utility” might do) of what we have taken to be “calculus” in all this time.
May we not be entitled to wonder whether, in the other – to me quite obscure – branches of their philosophy, in their philosophy of history, of economics, of sociology, Marx and Engels might have been equally ignorant, equally arrogant, and equally incomprehensible? I, myself, have found Marx merely incomprehensible – not that I ever read deeply in his works. Well, not merely incomprehensible; I have found him irritating as well, especially in those sarcastic footnotes in Das Kapital, which gave license to all the communists I have ever known, to be sarcastic in the same way, when speaking of their opponents. Sarcasm comes easier than careful exposition, as can be seen in the “these gentlemen” remarks of Engels below.
* * ****************************************************** * *
Friedrich Engels writing to Karl Marx (18 August 1881):
......Thus yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and was pleased to find that I did not need them. I compliment you on this feat. The thing is as clear as daylight, so that we cannot wonder enough at how stubbornly mathematicians insist on mystifying it. But this results from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and without circumlocution, does not enter their skulls. And indeed it is clear that dy/dx can be the pure expression of an ongoing process in x and y only if the very last traces of the quanta x and y vanish, leaving behind only the expression of the previous variation process, without any actual quantities remaining.
You need have no fear that in all this you will have been anticipated by any mathematician. This method of differentiation is so much simpler than any other that...[Raimi note: Alas, my German was simply not up to the translation of the rest of that passage.] . . . This procedure has the most surprising consequence, besides, in that it makes clear that the usual method, with its "leaving behind" of dx, dy, etc., is absolutely false. And this is the special beauty of the thing: only if dy/dx = 0/0, only then is the mathematical operation absolutely correct.
Old man Hegel had therefore been quite correct in his conjecture, that differentiation has at its foundation the requirement that the two variables be of differing powers, and at least one of them of power at least 2 or 1/2. Now we also understand why.
[Raimi note: I think this paragraph has been garbled by me in the translation; but then, I don't know what old Hegel had said, and was, when first reading this letter, surprised to learn that he had discussed this subject at all. Engel’s letter goes on as follows, and this part is contained in the Struik paper:]
When we say, in y = f(x), that x and y are variable, then so long as we stop there the proposition has no consequences, and x and y are still, pro tempore, nothing but constants. Beginning only when they actually, within the functional relationship, vary, do they in fact become variable; and only then can their relationship, previously hidden in the original equation, be brought out into the light of day, not as the quantities themselves but in their variability. The first difference quotient Δy/Δx exhibits this relationship as it flows from the actual variation, that is, as it results from a given variation; the final relationship dy/dx shows it in its universality, pure, and thus we see that the same dy/dx results from the various choices of Δy/Δx, though they themselves may differ according to the case. To arrive at the general relationship from the various cases, these cases must be liberated from being special cases as such. Following, therefore, the functional process of moving from x to x', with all its consequences, we can quietly let x' become x again; this is not merely the old variable named x any more, it has gone through an actual variation, and the result of the variation remains, even though we remove the thing itself.
Finally it becomes clear, as many mathematicians have long maintained without any rational grounds for believing it, that the differential quotient is the original relation, whose differentials are dx and dy: that the relationship of the formula itself requires that the two so-called irrational factors initially form one side of the equation, and only when goes back to the equation in its first form dy/dx = f(x) can one understand what to make of this, to be free of irrationals, and to set its rational expression in its place.
These thoughts have so seized me that they not only have been running around in my head all day, but I even had a dream the previous night that I had buttonholed a pal [?RAR] and gone through this whole differential matter with him, too.
Yours, F. E.
Ralph A. Raimi
Revised 26 June 2005
 Powell, A.B. and Frankenstein, M, Ethnomathematics, State University of New York Press, Albany, N.Y., 1997. Struik’s paper as it appears there is Chapter 8, Marx and Mathematics.
 I am sure this paragraph has been garbled by me in the translation; but then, I don't
know what old man Hegel had said, and was in 1996 surprised to learn that he discussed this subject at all. Struik’s paper says more about this than I do here.
 The italicized part of this last paragraph is omitted from Struik’s rendition, though he translates the immediately preceding part somewhat differently from what I have, apparently in an effort to clarify what Engels must mean. I found the last part, here italicized, quite obscure, especially the equation f(x)=dy/dx, which must be a misprint of some sort, perhaps in the printed version of the original letters.. The last (non-italicized) part of this paragraph is quoted by Struik in another part of his paper, while his quotation of the earlier part of this letter merely stops above the part I italicized here.
 My attempted translation (1996) of the last part of this quotation was quite wrong, I have now discovered. Dirk Struik translated it as follows: “… but that also last week in a dream I gave a fellow my shirt buttons to differentiate and this fellow ran away with them [und dieser mir damit durchbrannte].” Struik’s translation, recognizing its idiomatic nature, parenthetically appended the original German, which I believe also could be translated as “upon which the guy ran out on me.” Engels’s German is rather lighthearted in spots, and Struik’s translations, while infinitely better than mine in the places where I didn’t understand the German at all, does tend to render Engels somewhat more solemn than Engels intended. Apart from his diplomatic elisions (cf. footnore (1)), Struik is careful to translate accurately, or at least as he understands the intent of the original when mathematical statements are in question.