**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__[1],
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 17^{th} 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 19^{th} 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**.[2]
*

* *

[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.[3]**

* *

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][4]
and gone through this whole differential matter with him, too.

Yours, F. E.

Ralph
A. Raimi

Revised
26 June 2005

[1] 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__.

[2] 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.

[3] 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.

[4] 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.