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Source: Van Heijenoort, Jean. “Friedrich Engels and Mathematics,” in *Selected Essays* (Napoli: Bibliopolis, 1985), pp. 123-151, article written in 1948.

Note: The list of references for the whole book as well as this article, not reproduced here, is given on pp.153-166.

Friedrich Engels has passed judgment on many points in mathematics and its philosophy. What are his opinions worth? Important in itself, this question has a more general interest, for Engels' views on mathematics are part of his 'dialectical materialism', and their examination gives a valuable insight into this doctrine.

Mathematics is such a special branch of intellectual life that a preliminary question must be asked of anyone who ventures, as a philosopher, to investigate its nature and its methods: exactly what does he know in mathematics? Although the answer to this question does not forthwith determine the value of the solutions offered by the philosopher, it is nevertheless and indispensable preparation for examining them.

The programs of the German schools in the 1830s as well as his own inclinations lead young Engels toward an education that was more literary than scientific. True enough, at the Elberfeld high school, which he leaves before he is seventeen, he attends classes in mathematics and physics, even with a satisfactory record, but they remain quite elementary, and the young student does not seem to take any special interest in them. What attracts him most is literature, languages and poetry. After the study of law has held his attention for a moment, he is soon learning how to become a business man, which does not prevent him from devoting his spare time—and he has plenty of it—to writing poetry, composing choral pieces, drawing caricatures. As an unsalaried clerk in the export business of Consul Heinrich Leopold in Bremen, no doubt he knows the elementary rules of arithmetic, but no document of that period—and Engels is going through years that are critical in the molding of a young man—shows that he has any interest in sciences in general and mathematics in particular. Engels soon passes from poetry to that hall-literary, half-social criticism that the censorship is then trying to keep within well defined bounds. His great man at the time is Ludwig Börne.

A new impulse to the intellectual development of the young
man comes from reading Strauss' book, *Das Leben Jesu, *whose first volume
came out in 1835. Engels soon abandons religion definitively. However, unlike
the eighteenth-century French *philosophes, *who, in their fight against
religion, leaned directly on natural sciences and knew them rather well, Strauss
takes as his point of departure the contradictions in the Scriptures, and young
Engels' break with religion does not immerse him in the great stream of sciences,
as so often happens.

Through Strauss Engels comes into contact with Hegel, who immediately enthralls him. He is nineteen years old. Unlike Marx, who had studied Greek philosophers, Descartes, Spinoza, Kant, Leibniz, Fichte before tackling Hegel, Engels plunges into the latter's books with hardly any philosophical background. With the encyclopedic character of Hegel's works, where there is an answer to everything, the result is that Engels soon sees many a problem through the spectacles of the master sorcerer. Many years later, when examining a question, he will first read what 'the old man' has written on the subject.

In 1869, at forty-nine years of age, Engels retires from business and goes to live in London. Then, he writes

I went through, to the extent it was possible for me, a complete 'moulting' in mathematics and the natural sciences, and spent the best part of eight years on it [1935, page 10].

A few lines below, he speaks of

my recapitulation of mathematics and the natural sciences [1935, page 11].

What is Engels' mathematical knowledge on the eve of this 'moulting'? No positive document exists that would establish what his interest in mathematics has been between his school years and 1869. A clear picture, however, springs out of the mass of biographical documents. Names of mathematicians and titles of mathematical works are absent from writings and letters where hundreds of names and titles belonging to many spheres of intellectual activity can be found. The correspondence between Marx and Engels is especially valuable in this respect, for it enables us to follow the activities of the two friends, their readings, the fluctuations of their interests, from week to week, at times from day to day. Now, here no more than in the other writings of Engels that precede the 'moulting' of 1869 is there any trace of special interest, or simply of any interest at all, in mathematics on Engels' part. When Marx touches a mathematical point, as for instance in his letter of May 31, 1873, where he speaks of his project of

mathematically determining the main laws of crises [Marx and Engels 1931, page 398],

Engels does not react.

The only scrap of information that we can glean on the subject
is that, in 1864, Engels read Louis Benjamin Francœur's *Traité
d'arithmétique, *published in Paris in 1845. This is an elementary
arithmetic book, for the use of bank clerks and tradesmen. The very fact that
Engels studies such a book and comments on it in a letter to Marx (on May 30,
1864; Engels' comments are trifling [Marx and Engels 1930, page 173]), while
he does not mention any other mathematical work during some thirty years, is
enough to gauge the level of his interest and knowledge in mathematics prior
to 1869.

It seems therefore established that, until the 'moulting'
of *1869, *Engels hardly possesses more than the rudiments of elementary
arithmetic. As for this 'moulting' itself, of what does it consist? In
sciences other than mathematics, for example in chemistry, physics or astronomy,
the list of books that Engels mentions is abundant enough to permit us to follow
his progress in these domains with satisfactory accuracy. But, in mathematics,
the list is rather poor. Engels reads much more, for example, in astronomy,
a rather special science at that time, than in the whole field of pure mathematics.
In fact, only one work of pure mathematics, as far as we can ascertain, was
ever studied by Engels, that of Bossut (see Engels 1935, pages 392 and 636).

Charles Bossut published his *Traité de calcul différentiel
et de calcul intégral *in the year VI (1798). It is clearly a minor
work of a minor mathematician. The book was never reprinted. Neither Larousse's
*Grand dictionnaire universel du XIXe *siècle, nor *La grande
encyclopédie, *nor Maximilien Marie's *Histoire des sciences mathématiques
et physiques *mention it in the rather lengthy list of Bossut's works. Marie
adds:

We shall say nothing of Bossut's didactic works: they have lived the length of time that works of that kind live, twenty or thirty years, after which, methods having changed, students must have recourse to new guides [1886, page 24].

This rather severe judgment was still too lenient in that case, for Bossut had followed Newton in his presentation of the principles of the infinitesimal calculus, and the treatise was published at the very time when Lagrange was introducing a new rigor in this field, so that Bossut had to add to the end of his introduction the following paragraph:

Citizen Lagrange has presented the metaphysics of the calculus under a new
light, in his *Théorie des fonctions algébriques*; but
I have obtained knowledge of this excellent work only after mine was completed
and even largely printed [1798, page lxxx].

How, eighty years later, can Engels take as his guide in a fundamental question a work already out-of-date while in press, and follow this guide precisely in the domain where it had become most obsolete? The only possible answer to this question is that Engels did not know nineteenth-century mathematics and was not interested in it, that he found Bossut's book by chance and that he had no qualms about using it because the out-of-date ideas of the author seemed, to his mind, to confirm his own conception of the infinitesimal calculus, inherited from Hegel.

Let us note that Engels' conception of the calculus is, as we shall. see, one of the keystones of his philosophical edifice, for there lies the 'dialectic' of mathematics. The importance of this question for his philosophical conceptions makes it still less justifiable for him to have followed in this domain so obsolete a guide as Bossut.

Engels appears to be as unfamiliar with the history of the infinitesimal calculus as with its principles. In a manuscript entitled 'Dialektik und Naturwissenschaft' (Dialectic and natural science) and written between 1873 and 1876, Engels mentions Leibniz as

the founder of the mathematics of the infinite, in face of whom the induction-loving
ass [*Induktionsesel*] Newton appears as a plagiarist and a corrupter [1935, page 603]. ^{[1]}

By the 'mathematics of the infinite' Engels understands, according to an eighteenth-century
expression, the infinitesimal calculus. His denunciation of Newton is, in a
coarsier [*sic*] language, a mere repetition of what can be found in Hegel, for
whom the invention of the calculus, falsely attributed to Newton by the English,
was exclusively due to Leibniz (see, for instance, Hegel 1836, page 451).

A few years later, in 1880, hence after more than ten years of 'moulting', Engels
wrote in the preface to *Dialektik der Natur *that the infinitesimal calculus
had been established

by Leibniz and perhaps Newton [1935, page 484].

We are still quite far from the truth.

Precisely on that question Engels could have used Bossut's work. The 'Discours préliminaire' in the first volume contains a history of the invention of the calculus which is one of the few good points of the book. Started at the beginning of the eighteenth century, the controversy about the priority of the invention was well-nigh settled when Bossut was writing in the last years of the century, so that he could conclude:

These two great men [Newton and Leibniz] have reached, by the strength of their geniuses, the same goal through different paths [1798, page li].

If the respective merits of Newton and Leibniz were clear to Bossut, the more so should they have been to Engels, writing eighty years later. But no, he has to repeat Hegel, on a point on which the philosopher is obviously wrong.

Let us consider the complex numbers, whose theory was completed in the nineteenth
century. It is an easy and, it seems, interesting subject for a man like Engels,
without training in mathematics. Three brief remarks are all we can find in
his writings. They show that, although Engels knows of the existence of the
complex numbers, he has never grasped their significance. In his book against
Dühring he sets complex numbers, 'the free creations and imaginations of
the mind' (1935, page 43), apart from other mathematical notions, which
are abstracted from the 'real world'. The same book contains a few sentences
on the square root of minus one,

which is, according to Engels,

not only a contradiction, but even an absurd contradiction, a real absurdity
[1935, page 125]. ^{[2]}

Finally, in an unpublished article, probably written in 1878 and entitled 'Die Naturforschung in der Geisterwelt' (Natural science in the world of spirits), he writes:

The ordinary metaphysical mathematicians boast with huge pride of the absolute irrefutability of the results of their science. Among these results, however, are the imaginary magnitudes, to which is thereby attributed a certain reality. When one has once become accustomed to ascribe to the [square root of] -1 or to the fourth dimension some kind of reality outside of our own heads, it is not a matter of much importance if one goes a step further and also accepts the spirit world of the mediums [1935, page 716].

These brief remarks reveal how little Engels understands what a complex number is, although these numbers were no longer a novelty at the time when he was writing. After a few precursors, Gauss had given in 1831 a geometric representation of complex numbers that removed from them the last trace of mystery. This representation had rapidly become current toward the middle of the century and, in 1855 for example, a quite elementary book could state:

It will probably be found, on a proper analysis, that the subject of imaginary expressions present no more difficulties than that of negative quantities, which is now so thoroughly settled as to leave nothing to be desired [Davies and Peck 1855, page 301].

Twenty years later, Engels is still stumbling over these 'thoroughly settled' difficulties.

Let us take another important development of mathematics in
the nineteenth century, non-Euclidean and n-dimensional geometries. After many
a futile attempt to prove Euclid's parallel postulate, mathematicians began
in the eighteenth century to wonder what its rejection would imply. Lobachevskii
presented the principles of a new geometry rejecting the postulate before the
departement of mathematics and physics of the Kazan
University in February 1826. But his lecture was not published and left no trace.
In 1829-30, he presented his new conceptions in a magazine printed by the same
university, but they did not immediately penetrate into the mathematical world,
owing to the remoteness and the language of the publication. In 1832 János
Bolyai, which had conceived a new geometry a few years earlier, independently
of Lobachevskii, published his famous *Appendix*. It then became known
that Gauss had been in possession of similar, but unpublished, results for quite
a few years.

Lobachevskii soon started publishing his works in French and
German, so that they were more easily read in Western Europe. The new ideas,
however, made slow headway until the middle of the century. Then comes Riemann's
probationary lecture, 'Ueber die Hypothesen, welche der Geometrie zu Grunde
liegen' (On the hypotheses that lie at the basis of geometry), on June 10, 1854.
Lobachevskii's *Pangéometrie, * published in French in Kazan in
1856, is translated into German in 1858 and into Italian in 1867. Bolyai's *Appendix
is * translated into French in 1872. Riemann's fundamental work, printed
in 1868, *is * translated into French in 1870 and, in 1873, published in
English in *Nature, * a magazine which, most likely, Engels is reading
regularly at that time. Gauss' unpublished manuscripts and private letters are
becoming known; some of his letters on non-Euclidean geometry are translated
into French in 1866. Helmholtz gives two lectures, in 1868 and 1870, on the
foundations of geometry. Beltrami's important work, showing for the first time
that non-Euclidean geometry has the same logical consistency as Euclidean geometry,
is published in 1868 and translated into French in 1869.

Riemann's probationary lecture of 1854 also marks a great
step forward for *n*-dimensional geometries. Grassmann's *Ausdehnungslehre,
* whose first edition dates from 1844, and Cayley's works beginning the same
year have already laid the foundations of the new theories. From then on, progress
is rapid. Cayley's epoch-making *A sixth memoir upon quantics* is published
in 1860 and an enlarged edition of the *Ausdehnungslehre* in 1862.

All these dates show that the year 1870 marks the time at
which the mathematical world becomes familiar with non-Euclidean and *n*-dimensional
geometries. At that date far-sighted pioneers have already begun to use the
new mathematical conceptions in other fields of science. As early as 1854 Riemann
suggests that some regions of our space might be non-Euclidean and that only
experience can decide. In 1870 Clifford develops the idea that Euclid's axioms
are not valid in small portions of our space and that

this variation of the curvature of space is what really happens in
that phenomenon which we call the *motion of matter* [1870, page 158].

From 1863 on, Mach attempts to apply the new geometries in physics and chemistry. After 1867 Helmholtz tries to connect the new ideas on the foundations of geometry to his researches in physiology.

The years [sic] 1870 also sees the beginning of the popularization
of the new conceptions. Helmholtz presents them before a group of non-mathematicians
in Heidelberg (in Helmholtz' works this lecture is always dated 1870; however,
in his 1876, Helmholtz himself says that the lecture was given in 1869). In
order to make himself understood he uses an illustration which will be repeated
in the innumerable works of popularization that are soon coming to light, that
of two-dimensional intelligent beings living and moving on a curved surface
and incapable of perceiving anything outside of this surface; their geometry
would be non-Euclidean. Let us notice that a slightly abridged version of this
popular exposition is published on February 12, 1870, in *The Academy, *
a magazine published in London, hence easily accessible to Engels. In 1876 Helmholtz
published an enlarged version of his lecture under the title 'Ueber den Ursprung
und die Bedeutung der geometrischen Axiome' (On the origin and significance
of geometrical axioms) in the third part of his *Populäre wissenschaftliche
Vorträge, * a book that a man like Engels, right in the middle of his
'moulting', can hardly ignore. Engels repeatedly quotes the second part of Helmholtz'
book in his writings of that period, hence he must have seen the third part.

From 1870 on, non-Euclidean and *n*-dimensional geometries
elicit general curiosity, somewhat like the theory of relativity at the end
of the First World War and nuclear fission at the end of the Second. The German
philosopher Hermann Lotze, by no means a mathematician, writing during these
very years, speaks of

the much talked about fourth dimension of space [...], which is now mooted on all sides [1879, pages 254-255].

Precisely at that time Engels is going through his scientific 'moulting'. However, he does not pay any attention to these developments. This is the more surprising since, firstly, the new mathematical conceptions have extremely important philosophical implications and, secondly, their study does not require very deep mathematical knowledge or technique. Helmholtz had already noted these two points in 1870:

It is a question which, as I think, may be made generally interesting to all who have studied even the elements of mathematics, and which, at the same time, is immediately connected with the highest problems regarding the nature of the human understanding [1870, page 128].

In brief, it is precisely the kind of question which, it seems, should enthrall a man like Engels, at that period of his intellectual life. His only mention of the subject, however, is in the article already quoted, 'Die Naturforschung in der Geisterwelt'.

Modern spiritualism, born in the United States toward the middle of the nineteenth century, bloomed in Europe shortly afterwards. In the 1870s interest in it was great and polemics about it numerous. Precisely the same years saw a widespread diffusion of the new mathematical theories. Zöllner, an astrophysicist in Leipzig, not without scientific talent, became converted to spiritualism and tried to explain spiritualistic phenomena by the fourth dimension. In his article Engels jeers at Zöllner, but, as much as at Zöllner, he jeers at the fourth dimension; he even jeers at established mathematical results. The article does not show the slightest effort at understanding the new mathematical developments and produces a very painful impression.

This article at least tells us that Engels knows of the existence of the new geometries. But all he does is practically to put them on the same plane as spiritualism. These upsetting and exciting ideas, destined to a great future, rich in philosophical implications, discussed at the time by everyone showing any interest in science, do not retain at all the attention of Engels, who simply scoffs at them. Such a strong resistance on his part to the new ideas can by no means be due to episodical causes. It has its roots in his own conception of mathematics. We shall soon understand why Engels' mind is closed to these questions. In the meantime, let us take note of the fact.

Let us sound out once more Engels' mathematical knowledge.
In notes for his *Dialektik der Natur, * commenting on the change of base
in the writing of numbers, he states that

All laws of numbers depend on, and are determined by, the system used [1935, page 671].

This is not true. Passing from one base to another merely changes the symbols representing the number, but by no means its arithmetical properties. For this false statement Engels gives an equally false example:

In every system with an odd base, the difference between even and odd numbers disappears [1935, page 671].

A number remains even or odd independently of the base used. It would not be without interest to show how Engels was led by his 'dialectic' to such a senseless affirmation, but suffice it to note here, in this study of Engels' mathematical knowledge, that all this is quite elementary arithmetic and would not puzzle an average sixteen years old student.

The picture emerging from this research is too dark and somewhat distorted, one may perhaps object. Truly enough, the argument would run, Engels does not pay much attention to pure mathematics during his 'moulting', but he reads quite a few books on astronomy and physics, where mathematics is used on every page, and he has an opportunity to become familiar with mathematical methods. This objection contains a grain of truth, but no more than a very tiny grain. Engels learned most of whatever he knew in mathematics from books on physics. This is clear, for example, from his oft-repeated assertion that the rules of the infinitesimal calculus are false from the viewpoint of physics; he never studied the mathematical theory that logically justifies the physicist's apparent approximation. But no more than the quality should the quantity of Engels' mathematical knowledge thus acquired be overestimated. A small incident will permit us to gauge it.

In the second preface to the *Anti-Dühring, * written
in September 1885, hence after many years of 'moulting', Engels states:

[ ... ] Hegel emphasized that Kepler, whom Germany let starve, is the real
founder of modern mechanics of heavenly bodies and that Newton's law of gravitation
is already contained in all three Kepler's laws, even explicitly in the third
one. What Hegel shows with a few simple equations in his *Naturphilosophie*,
§ 270 and additions (Hegel's *Werke, * 1842, volume VII, pages 98
and 113-115), appears again as a result of modern mathematical mechanics in
Gustav Kirchhoff's *Vorlesungen über mathematische Physik, * 2nd
edition, Leipzig, 1877, page 10, and in a mathematical form which is essentially
the same as the simple one first developed by Hegel [1935, pages 11-12].

Let us open the two books mentioned by Engels at the pages he indicates. In Kirchhoff's book we do find the derivation of Newton's law of attraction from Kepler's three laws, as it can still be found in any elementary textbook of mechanics. It requires two or three pages and makes use of the integral

calculus and elementary differential equations. Now, in Hegel we read something much shorter:

In Kepler's third law, A^{3}/T^{2}
is the constant. Let *us * write it A.A.^{2}/T^{2}
and, following Newton, let us call A/T^{2}
the universal gravitation; then the expression of the action of this so-called
attraction is inversely proportional to the square of the distance [1842,
pages 98-99].

In these puerile lines, Hegel does not see, among other things, that the variable distance between the planet and the sun is not the semimajor axis of the eliptic orbit. On page 115, also mentioned by Engels, the same error, with a few others added for good measure, is repeated. Hegel's greatness rests on other achievements than these absurdities dictated by a deep-rooted and violent prejudice against the Englishman Newton as well as by an inveterate lack of understanding of mathematical methods.

Half a century later, after many years of personal 'moulting', with the correct derivation under his eyes in Kirchhoff s book, Engels does not see Hegel's mistakes. Much worse, he states that the two derivations are 'essentially the same'. No, indeed, we cannot say that Engels learned much more mathematics from physics books than from mathematical treatises.

What should we retain from all this? Engels does not show the slightest aptitude for mathematics; he does not know any of its developments in the nineteenth century; his judgments in the philosophy of mathematics are based on conceptions prevalent ninety or a hundred years before the time he was writing, while this interval had seen tumultuous and far-reaching progress; even so far as eighteenth century mathematics is concerned, he never comes into intimate contact with it; he only knows its problems through Hegel, a rather poor guide in that domain. Nevertheless, as we shall see now, Engels does not hesitate to pronounce sweeping judgments on mathematics and its philosophy.

Engels' conception of mathematics matches well his epistemology,
the copy theory of truth, and even forms its crudest part. As, in general, ideas
are for him nothing but 'mirror images' ^{[3]} of material things, mathematical concepts in particular are nothing but 'imprints of reality' (1935, page 608).

The first consequence of such a theory is to confuse what is mathematical and what is physical; mathematics is no longer anything more than a branch of physics. That Engels does not shrink from such an implication is shown beyond question by his writings.

In order to give examples of undoubtedly true propositions, he mentions those which state

that 2 X 2 = 4 or that the attraction of matter increases and decreases according to the square of the distance [1935, page 496].

Engels does not hesitate to put on the same plane a mathematical theorem and a physical law. History has come to deride his conception: experience has compelled us to abandon Newton's law and adopt another theory, while we cannot see how experience could force us to question a numerical statement. This clearly shows the difference in nature between the two propositions mentioned.

As examples of

eternal truths, definitive, ultimate truths [1935, page 91],

Engels mentions

Here again mathematical theorems are intermingled with empirical observations. For Engels the proposition that the sum of the three angles of a triangle is equal to two right angles has the same kind of truth as the empirical statement that Paris is in France. He writes this in 1877, when it is already widely recognized that the first proposition follows from a certain set of axioms, namely those of Euclidean geometry, and will perhaps not follow from some other set of axioms. But we have seen how obstinately Engels keeps his eyes closed to non-Euclidean geometries. They are too great a threat to his identification of mathematics with physics.

According to Engels, mathematical concepts are

taken from nowhere else than from the real world [1935, page 43].

They are

exclusively borrowed from the outside world, not sprung from pure thought in the head [1935, page 93].

Let us note the word 'exclusively'. That experience has elicited certain mathematical notions is indisputable. But it has by no means directly imprinted them on a passive human brain. Looking at a spider's thread or at a stretch of still water, never will a man conceive the mathematical straight line or plane without an intellectual activity irreducible to mere observation, to mere 'mirroring'. As for more complex mathematical concepts, it is soon impossible to tell from which natural objects they would be the 'mirror images'. Yes, the mathematician receives many suggestions from experience; but the quid proprium of mathematics is to pass to the limit, to deal with perfect objects, lines without breath, surfaces without thickness, and to deal with them not by means of observation, but of logical reasoning.

To take an example, let us consider the number [pi], the ratio of the circumference of a circle to its diameter. If [pi] were simply given by experience, we would have to build a wheel of metal and measure with the greatest possible accuracy its circumference and its diameter. Their ratio would give [pi], or rather an approximation of [pi]. However, the mathematician can, by pure reasoning, compute a mathematical [pi] with an unlimited precision. He can make statements about this mathematical [pi]— for example, that it is an irrational, transcendental number—that would be meaningless for the physical [pi]. In Engels' writings there is no indication that he would draw any distinction between the two concepts; more accurately, for him, the mathematical [pi] would disappear behind the physical [pi].

For Engels the share of experience in the formation of mathematical concepts is much more than mere suggesting. He writes:

Pure mathematics has for its object the spatial forms and quantitative relations of the actual world, hence a very real stuff [1935, page 43].

Mathematics, as a human creation, is obviously part of 'reality'. If Engels wanted to say nothing more than that, it would be a platitude. However, what he understands by 'actual world' is nature, the physical, material world, and his statement is false, for it is by no means accurate to say that mathematics has for its object only the relations of the physical world. The same false conception is repeated again and again:

The results of geometry are nothing but the natural properties of the different lines, surfaces and bodies, or of their combinations, that in great part already appeared in nature long before men existed (radiolaria, insects, crystals, and so on) [1935, page 393].

That a shell has the shape of a certain mathematical curve may be of great interest to the biologist and suggest, for example, an exponential growth, but it is of no great consequence for the mathematician. Firstly, the mathematical curve is not an 'imprint' of the shell upon the mathematician's brain; it is defined in mathematical terms. Secondly, the mathematician will never prove theorems about the curve by measuring the shell. What he could at most expect is to receive some suggestion from experience; his real task would then only begin, and he could fulfill it only by axiomatically deriving new propositions about the curve from its definition and already known theorems. The mathematician may even decide to take as his point of departure assumptions that are not 'relations of the actual world', that are not 'natural properties' of insects or crystals, and build geometries that transcend our experience. In a study of Engels' philosophy, Sidney Hook has already noted (1937, page 261)

the curious reluctance on the part of orthodox Hegelians and dialectical
materialists to admit that hypotheticals contrary to fact,* i. e*. judgments
which take the form '*if* a thing or event had been different from what
it was', are meaningful assumptions in science or history.

Nowhere is this tendency more apparent than in Engels' attitude toward mathematics, and nowhere is it more dangerous. It does away with the if-then aspect of mathematics.

One of Engels' most surprising writings is a note written in 1877 or 1878 and entitled 'Ueber die Urbilder des mathematischen "Unendlichen" in der wirklichen Welt' (On the prototypes of the mathematical 'infinite' in the real world). It would be a tedious and not too rewarding task to unravel the skein of exaggerations, misunderstandings and plain mistakes contained in these few pages. The core of it is that Engels undertakes to show that every mathematical operation is 'performed by nature'; nature differentiates, integrates, solves differential equations exactly like the mathematician. Both sets of operations are 'literally' (1935, page 467) the same, except that

the one is consciously carried out by the human brain, while the other is unconsciously carried out by nature [1935, page 467].

For instance, the molecule is a differential, and

nature operates with these differentials, the molecules, in exactly the same way and according to the same laws as mathematics does with its abstract differentials [1935, page 466].

Let us not tarry in investigating what this nature operating with human laws is, let us see how Engels justifies this animistic view. He offers the example of a cube of sulphur immersed in an atmosphere of sulphur vapor in such a way that a layer of sulphur, the thickness of a single molecule (the differential!), is deposited in three adjacent faces of the cube. But even with this artificial example, custom-built to prove (!) a universal law, Engels entangles himself and must finally note the discrepancy between the physical process and the mathematical reasoning. He tries to explain it away in a short sentence, saying that,

as everyone knows, lines without thickness or breath do not occur by themselves in nature, hence also the mathematical abstractions have unrestricted validity only in pure mathematics [1935, page, 467].

This is precisely the point at issue, which Engels refuses to confront openly, but must surreptitiously concede. Now, where is the 'literal' identity of a physical process with a mathematical reasoning?

According to Engels, mathematics makes use of only two axioms:Mathematical axioms are expressions of the most indigent thought content, which mathematics is obliged to borrow from logic. They can be reduced to two:

1. The whole is greater than the part. This proposition is a pure tautology
[...]. This tautology can even in a way be *proved * by saying: a whole
is that which consists of many parts; a part is that of which many make a
whole; therefore the part is less than the whole [...].

2. If two magnitudes are equal to a third, then they are equal to one another. This proposition, as Hegel has already shown, is an inference, the correctness of which is guaranteed by logic, and which is therefore proved, although outside of pure mathematics. The other axioms about equality and inequality are merely logical extensions of this conclusion.

These meager propositions could not cut much ice, either in mathematics or anywhere else. In order to get any further, we are obliged to introduce real relations, relations and spatial forms which are taken from real bodies. The notions of lines, surfaces, angles, polygons, cubes, spheres, and so on, are all taken from reality [1935, page 44-45].

This passage shows that, by an axiom, Engels does not at all
understand the same thing as mathematicians do. Firstly, he undertakes to 'prove'
his two axioms (one of which, by the way, is a 'tautology'!). Secondly, these
two arbitrarily selected propositions are insufficient as points of departure
for mathematics. ^{[4]} Mathematicians need quite a few more assumptions on sets,
numbers, points, lines, and so on. Engels would not deny this. In fact, these
are the 'relations' that he mentions in the last paragraph of the passage quoted
above. These propositions are, for him, directly taken from physical reality
and are, therefore, 'materially' true. The idea that mathematicians can successively
adopt contradictory sets of axioms and ascertain what each set implies is thoroughly
alien to him.

Engels' conception of mathematical axioms as immediately given by the physical world leads him to reject the deductive method of proof used in mathematics. In a note written during the preparation of his book against Dühring we find the following lines:

Comical confusion of the mathematical operations, which are susceptible
of material demonstration, susceptible of being tested, because they rest
on immediate material, although abstract, observation, with the *purely*
logical operations, which are only susceptible of a deductive demonstration,
hence incapable of having the positive certitude that the mathematical operations
have,—and how many of these [logical operations] are even false! [1935,
pages 394-395].

It is all topsy-turvy. Engels draws a vaguely correct distinction between factual observation and logical deduction; but, then, he puts mathematical proof on the side of material observation. His statements those quoted and quite a few others of the same sort—are nothing less than a negation of mathematics, a destruction of the structure started with Greek geometry and raised to such heights in the last two hundred years. Without the cement of logical deduction, mathematics would be reduced to a kind of land surveying, made up of empirical recipes, haphazard observations and strange coincidences. The position seems indeed untenable. But Engels' words are clear, and they do not lack self-assurance.

In the discussion on the part of physical experience in mathematics, three points are involded: the nature of axioms, the deductive method, the origin of fundamental concepts.

The nature of mathematical axioms, whether they are a priori truths or generalizations from observations, was a live subject of discussion up to the middle of the nineteenth century. After the appearance of non-Euclidean geometries and other mathematical developments, the question became fairly settled for everybody well enough informed. Axioms are assumptions, whose 'truth' is irrelevant and, in a sense, meaningless in the field of mathematics. It is up to the physicist to decide which set of axioms should be used in the study of nature, but this choice is not a mathematical question anymore. There are perhaps limits to the if-then conception of mathematics. One could claim that the sequence of natural numbers is directly given to us by an intuition that is prior to, and independent of, the selection of any axiom system, and, besides, that the very notion of axiom system already involves that of natural number. Beyond the various set theories, there is perhaps an 'absolute' universe of sets. And, finally, the logic that takes us from 'if' to 'then' cannot itself be relativized. On each of these points there are arguments and counterarguments.

We do not intend to enter this controversy here. Our aim is simply to delimit the area of discussion and to show that Engels' opinions are well outside the range of those of competent workers in the field since the middle of nineteenth century. In mathematics there is simply no question of proofs based on physical measurements, of definitions directly 'imprinted' by the physical world, of axioms that are nothing but physical laws.

Engels' conception of mathematics is a crude form of empiricism. It bears a certain resemblance to the conceptions of two of its contemporaries, Herbert Spencer and John Stuart Mill. These two philosophers, however, are much more aware of the difficulties of their positions, make painstaking efforts to answer all possible objections and carefully qualify their statements. Engels makes sweeping assertions and jeers at those who do not think like him. On one point only does he try to strengthen his theses. His conception of ready-made mathematical notions directly taken from the physical world is so contrary to the actual development of knowledge that he has to mitigate it by an idea avowedly borrowed from Spencer, the acquisition of mathematical axioms through heredity (Engels' epigones prefer not to mention this influence):

By recognizing the inheritance of acquired characters, it [modern science] extends the subject of experience from the individual to the genus; the single individual that must have experienced is no longer necessary, its individual experience can be replaced to a certain extent by the results of the experiences of a series of its ancestors. If, for instance, among us the mathematical axioms seem self-evident to every eight years old child, and in no need of proof from experience, this is solely the result of 'accumulated inheritance'. It would be difficult to inculcate them by proof upon a Bushman or Australian Negro [1935, pages 464-465].

The same idea is repeated elsewhere in almost identical terms:Self-evidence, for instance, of the mathematical axioms for Europeans, certainly not for Bushmen and Australian Negroes [1935, page 385].

We finally learn the source of the idea:Spencer is right inasmuch as what thus appears to us to be the *self-evidence
* of these axioms is* inherited* [1935, page 608].

It is sufficient to try to state precisely Engels' conception
to see how empty it is. What experience is inherited? Is it our familiarity
with solid objects, our 'converse with things', to use Spencer's expression?
In this respect, however, non-whites are not inferior to whites, unless we assume
that they have not existed as men as long a time, that is, that they are much
closer to the ape; but this is a vulgar assumption lacking any scientific basis.
Or shall we accept, as the other possible interpretation, that mathematical
axioms have become obvious to white children by heredity during the few centuries
that they have been regularly going to school? Certainly no difference between
white and non-white children has yet been ascertained in grasping the evidence
of mathematical axioms. And certainly this unobserved difference cannot be invoked
in order to explain the 'proof through experience' ('Erfahrungsbeweis') of mathematical
axioms. Let us say no more on that. ^{[5]}

The relation that the mathematics of variable magnitudes has to the mathematics of constant magnitudes is on the whole the relation of dialectical to metaphysical thought [1935, pages 125-126].

The two domains in which mathematics are split are logically irreconcilable. What is true in one is false in the other:

With the introduction of variable magnitudes and the extension of their variability to the infinitely small and the infinitely large, mathematics, otherwise so austere, has committed the original sin; it ate of the tree of knowledge, which opened up to it the career of the most gigantic achievements, but also of errors [1935, pages 91-92].

Or:

[...] higher mathematics, which [...] often [ ...] puts forward propositions which appear sheer nonsense to the lower mathematician [1935, page 602].

Or:

Almost all the proofs of higher mathematics, from the first proofs of the differential calculus on, are false, strictly speaking, from the standpoint of elementary mathematics [1935, page 138].

Not only are the proofs false, they simply do not exist:

Most people differentiate and integrate not because they understand what they are doing, but by pure faith, because up to now it has always come out right [1935, page 92].

Engels himself dimly feels how rash his statement is and tries to mitigate it with the words 'most people'. But what does he mean by that? Do the theorems of the infinitesimal calculus have proofs or not? If they do, then Engels' whole structure collapses, and he merely says that some people who use the calculus do not know or do not remember the derivation of the rules they use; such a situation is, of course, not confined to the calculus or even to mathematics; whether the people ignorant of the proofs of the rules are few or many, this has nothing to do with the point at issue, so long as the proofs exist. Or do the proofs perhaps not exist? In which case Engels should not speak of 'most people', but of everybody using the calculus without proofs. He was apparently ill at ease about making such a statement, and, by speaking of 'most people', he tried to cover its silliness with a fog of ambiguity.

The idea that emerges from this confusion is that the mathematician or the physicist, when using the calculus, does not follow the rules of logic, elementary geometry and arithmetic. Engels apparently has in mind the replacing of the increment of a function by its differential. When establishing a differential equation, the physicist often reasons as if a small segment of the curve were straight, that is, as if a function were linear in a small interval; but he knows that the step is perfectly justified by passing to the limit. He could obtain the same result in a strictly logical way by using the law of the mean; the procedure would be somewhat longer; he used it a few times when he learned the calculus, and convinced himself that he could use a method of approximation, which is a timesaving device, but does not in any way shake the logical foundations of the calculus.

True enough, when the infinitesimal calculus came into general use, in the eighteenth century, confusion reigned on that point, and many mathematicians were more concerned with obtaining new results than with strictly justifying their proofs. Such a situation, however, was very unsatisfactory and great efforts were soon spent to establish the calculus on a logically firm basis. Between 1820 and 1830, fifty years before the time Engels was writing, Cauchy gave a definition of the derivative as a limit, and the difficulty against which Engels is stumbling, namely the definition of differentials, disappeared:

In the mathematical analysis of the seventeenth and most of the eighteenth
centuries, the Greek ideal of clear and rigorous reasoning seemed to have
been discarded. 'Intuition' and 'instinct' replaced reason in many important
instances. This only encouraged an uncritical belief in the superhuman power
of the new methods. It was generally thought that a clear presentation of
the results of the calculus was not only unnecessary but impossible. Had not
the new science been in the hands of a small group of extremely competent
men, serious errors and even debacle might have resulted. These pioneers were
guided by a strong instinctive feeling that kept them from going far astray.
But when the French revolution opened the way to an immense extension of higher
learning, when increasingly large numbers of men wished to participate in
scientific activity, the critical revision of the new analysis could no longer
be postponed. This challenge was successfully met in the nineteenth century,
and today the calculus can be taught without a trace of mystery and with complete
rigor [*Courant and Robbins* 1948, page 399].

For Engels, the history of mathematics followed exactly the opposite direction. Speaking of the derivative, he writes:

I mention only in passing that this ratio [the derivative] between two vanished quantities [...] is a contradiction; but that cannot disturb us any more than it has disturbed mathematics in general for almost two hundred years [1935, page 141].

Mathematics has indeed been disturbed by the 'contradiction', had spent great efforts in order to overcome it and had, by Engels' time, succeeded. But Engels paints a truly fantastic picture of the development of science. For him, the eighteenth century had known a 'metaphysical' science, meaning that scientists were then following logic, operating with 'fixed categories' and ignoring change. In the nineteenth century science had become 'dialectical', that is, had accepted contradictions as a token of truth. He presents this picture many times in his writings, and it is interesting to see what part mathematics plays in it. According to Engels, 'higher mathematics', that is, chiefly the infinitesimal calculus, is full of 'contradictions'; mathematicians have been forced to accept these contradictions, and their science is pure absurdity from the standpoint of logic. Then, this science has induced other sciences also to accept contradictions and had led them from the 'metaphysical' era of the eighteenth century to the 'dialectical' era of the nineteenth century:

Until the end of the last century, even until 1830, natural scientists were quite satisfied with the old metaphysics, because the real science did not go beyond mechanics, terrestrial and cosmical. Nevertheless, confusion was already introduced by the higher mathematics, which considers the eternal truth of the lower mathematics as a superseded standpoint, often affirms the contrary [of what lower mathematics does] and establishes propositions that appear to the lower mathematician as sheer nonsense. The fixed categories were here dissolving themselves, mathematics had entered upon a ground where even such simple questions as those of the mere abstract quantity, the bad infinite, were taking on a completely dialectical shape and forcing the mathematicians, against their will and without their knowledge, to become dialectical. Nothing more comical than the wriggles, the foul tricks and the makeshifts used by the mathematicians for solving that contradiction, for reconciling higher and lower mathematics, for making clear to their mind that that which appeared to them as an incontrovertible result was not pure idiocy, and in general for rationally explaining the point of departure, the method and the result of the mathematics of the infinite [1935, page 602].

By the 'mathematics of the infinite' Engels means, as we have seen, the infinitesimal calculus, and his conception can hardly be more incorrect. In the eighteenth century mathematics had acquired a great wealth of new results, without always bothering too much about strict proofs. In the nineteenth century, on the contrary, the accent was on rigor, and very strict standards of logic were followed. Great progress was made in that direction, and among what Engels calls 'the wriggles, the foul tricks and the makeshifts' of the mathematicians are some of greatest achievements of the human mind. The very year 1830, which he gives as the line of demarcation between 'metaphysics' and 'dialectic' in science, marks, with Cauchy, the introduction of a new rigor in mathematics. Engels' picture is the exact opposite of the actual historical development.

If Engels still considers the calculus to be irreducible to
logic, it is because, one might say, he does not know the nineteenth century
developments in that field. True enough. We have seen that his source of information
on the subject was Bossut's treatise, which belongs, not only by the date of
its publication, but also by its spirit, to the eighteenth century. However,
lack of information cannot absolve Engels. Firstly, in any case *ignorantia
non est argumentum * and, secondly, in the present case we must ask the question:
why did Engels not study these nineteenth-century developments? After all, he
presented his wrong conception of the calculus in his book against Dühring,
which was published in the last quarter of the nineteenth century. Could he
not have paid attention to what mathematicians had done in the first three quarters
of that century?

A complete answer to this question would lead us into an examination of Engels' ways of thinking, writing and polemizing. We would have to show by many other examples how he often disregards facts when they do not suit him, how he fads to mention and refute possible objections to his blunt statements, how he answers an opponent by a joke or by calling him names. Suffice it to say here that Engels believed he had found in the conceptions of the calculus temporarily prevalent in the eighteenth century a confirmation of the ideas remaining in his own mind since he had read Hegel, and he simply did not bother to investigate any further.

Even if Engels had not followed the mathematical developments
that occurred in the thirty or fifty years before the time he was writing on
mathematics, he could have found a better guide than Bossut; he could have used,
for instance, Lacroix's treatises, the complete one published in 1797 or the
elementary one published in 1802; these works are far superior to Bossut's;
they became standard textbooks and ran into numerous editions up to the very
end of the nineteenth century. Although Lacroix was writing before Cauchy's
decisive contribution and had not yet a strict definition of the limit of a
function, his treatment is modern in spirit and, at the turn of the century,
he already defined the differential as the linear part of the increment of the
function, which is the present definition and could have dispelled many of Engels'
dark clouds of confusion. For that matter, Engels could also have read d'Alembert's
article 'Differentiel' in the *Encyclopédie, * dating from the middle
of the eighteenth century; d'Alembert still uses the intuitive notion of limit,
but his concise, clear and sagacious notice is a torch whose light could have
been most helpful to Engels more than hundred and twenty years later.

Engels, however, kept his eyes closed to the actual development of mathematics. His eyes are still closed when he undertakes to show how mathematics is full of contradictions. He does not hesitate to write that

one of the main principles of higher mathematics is the contradiction that in certain circumstances straight lines and curves are the same [1935, page 125].

This is apparently a reference to the calculus, and we have already seen what this 'contradiction' really is. The next one is simply whimsical:

[Higher mathematics] also establishes this other contradiction that lines which intersect each other before our eyes nevertheless, only five or six centimeters from their point of intersection, should be taken as parallel, as if they would never meet even if extended to infinity [1935, page 125].

It is not easy to see what Engels means here. Is it again the question of approximation in calculus? Is it an allusion to the fact that mathematicians can use a badly drawn figure for a correct proof? Anyway, these five or six centimeters have nothing to do with mathematics, and there is no contradiction here either. Engels finds that even 'elementary mathematics' is 'teeming with contradictions' (1935, page 125):

It is for example a contradiction that a root of A may be a power of A, and
yet A^{1/2} = [square root of] A [1935, page
125].

Whoever has studied the question of fractional exponents will have difficulty in finding a contradiction here. The proof given to young students consists precisely in showing that there is no contradiction in treating radicals as powers with fractional exponents and that it is, therefore, legitimate to extend the concept of power. This generalized power subsumes the power in the elementary sense of the word as well as the radical. Using an analogy, we could reconstruct Engels' thought thus: 'A cat is a feline; a tiger is a feline; hence a cat is a tiger. Here is a contradiction!' Old sophism. Why does Engels make this mistake? Probably because he considers contradictions to be the highest product of thought, mirroring 'motion', 'life' (see, for instance, 1935, page 124). Non-contradictory thought is for him hardly possible. Hence he has to discover contradictions everywhere. And he does! After roots come complex numbers:

It *is * a contradiction that a negative magnitude should be the square
of anything, for every negative magnitude multiplied by itself gives a positive
square [1935, page 125].

If one carefully rereads this sentence, it is simply impossible to find in it the contradiction imagined by Engels. The square of a negative number is a positive number; hence a negative number is not the square of a negative number. But why can it not be the square of some other kind of number? Where is the contradiction?

'Dialectic' manifests itself in mathematics not only by contradictions, but also by the law of the negation of the negation, whose validity Engels undertakes to prove by exhibiting examples. Here is the first:

Let us take an arbitrary algebraic magnitude, namely *a. * Let us negate
it , then we have *-a * (minus *a). * Let us negate this negation
by multiplying *-a * by -*a*, then we have +*a*, that is the
original positive magnitude, but to a higher degree, namely to the second
power [1935, pages 388-389].

Now comes a second example:

Still more strikingly does the negation of the negation appear in higher
analysis, [ ... ] in the differential and integral calculus. How are these
operations performed? In a given problem, for example, I have the variable
magnitudes *x* and *y* [ ...]. I differentiate* x * and *y*
[ ...]. What have I done but negate *x* and *y* [ ... ]? In place
of *x* and *y*, therefore, I have their negation, *dx* and
*dy*, in the formulas or equations before me. I continue then to operate
with these formulas and, at a certain point, *I negate the negation, *
that is, I integrate the differential formula [1935, pages 140-141; see also
page 392 and the footnote on page 388].

In these two examples 'to negate' means four different operations:
(1) to multiply by - 1, (2) to square a negative number, (3) to differentiate,
(4) to integrate. What is the common feature of these operations that would
allow Engels to subsume them under the concept of negation? A few pages later
he tells us that 'in the infinitesimal calculus it is negated otherwise than
in the formation of positive powers from negative roots' (1935, page 145). But
he never gives us the slightest hint as to what distinguishes the four 'negating'
operations from other mathematical operations. Or can any mathematical operation
be considered as a 'negation'? Then, what does the 'negation of the negation'
mean? It is both impossible and useless to criticize Engels' use of this formless
notion in the field of mathematics. *Quod gratis asseritur gratis negatur.
* Let us simply note that there is no mathematical rule or principle that
could possibly be, even by the farthest stretch of the imagination, identified
with Engels' negation of the negation.

After having witnessed the contempt with which Engels treats logic, we would never expect to read in his book against Dühring the following lines:

[ ...] formal logic is above all a method of arriving at new results, of advancing from the known to the unknown [1935, page 138].

Let us notice the words 'above all'. Formal logic is now for
Engels an *ars inveniendi, * a conception hardly dreamed of in the
heyday of Scholasticism. In fact, formal logic hardly is a method of discovery
in mathematics; imagination and intuition fulfill that role. In other sciences
it is still, if possible, more sterile for discovery. Why did Engels allow himself
such a blunder? The end of the sentence gives the answer:

[ ... ] and dialectic is the same, only in a much more eminent sense [1935, page 138].

Engels bestows such an extraordinary worth upon formal logic (which, poor soul, had never asked for anything like it!) only in order to ascribe it the more easily to his 'dialectic', to a much higher degree.

If we leave aside this last sleight of hand, Engels' main idea is that mathematics is divided into two incompatible domains and that the results of 'higher' mathematics, mainly the infinitesimal calculus, cannot be justified before the instance of 'lower' mathematics and formal logic. As we soon learn that 'lower' mathematics itself 'teems with contradictions', the whole edifice becomes quite shaky and, once we have seen what the 'contradictions' or the 'negation of the negation' actually are, not much remains.

These ideas have been inspired, of course, by Hegel. The second
section of the first book of his *Wissenschaft der Logik is *devoted to
Quantity and contains long passages on number, infinity and the infinitesimal
calculus. Hegels' remarks on this last subject are often interesting, especially
if we do not forget that they were written before 1812, at a time when the question
was not yet settled for mathematicians. Hegel, moreover, has up-to-date information;
for example, he mentions Carnot and extensively deals with Lagrange's work.
Hegel's remarks also show an effort to understand, which is absent from Engels'
writings. Finally, these remarks are embedded in a broad philosophical conception
that gives them scope and depth. In Engels everything is reduced to two or three
dry formulas on 'contradiction' or 'negation of the negation', which he hopelessly
tries to apply here and there.

It is true that behind *some *of Engels' contradictions
there are real problems, like the arithmetization of the continuum or the relation
between potential and actual infinite. These problems have preoccupied many
thinkers, from the Greeks to Kant, from Kant to the modern mathematicians. They
are at the bottom of still unsettled differences in the foundations of mathematics.
Engels sets himself to deal with Kant's antinomies, soon announces that

the thing itself can be solved very simply [1935, page 54],

and gives a few pages of explanations. Engels' solution is not too clear, but, so far as one can make out, coincides with what we have already seen above about the existence and role of contradictions in mathematics: the more, the better. According to Engels,

The infinite is a contradiction, and is full of contradictions. It is already
a contradiction that an infinity should be made up of mere finite parts, and
that is the case nevertheless [ ... ]. Every attempt to overcome these contradictions
leads [ ... ] to new and worse contradictions. Precisely *because * the
infinite is a contradiction, it is an infinite process, unwinding itself without
end in time and space. The overcoming of the contradiction would be the end
of the infinite [1935, page 56].

In these lines the words 'contradiction' and 'infinite' alternate without producing much light. Meanwhile, nineteenth-century mathematicians, men like Bolzano and Cantor, had attacked the problem and were making great progress. The only thing that can be said for Engels is that he occupies himself with an important problem, but nothing more; it cannot be said that he brings any appreciable contribution to its clarification. On the contrary, exactly as in the case of the infinitesimal calculus, Engels looks for a solution in a direction opposite to the actual development of science.

If we cannot claim to have dealt with every statement of Engels
on mathematics, an examination of those left out would not change, but rather
confirm, the conclusions emerging from our study of Engels' writings. ^{[6]} Some,
however, may challenge these conclusions on the ground that some of the quotations
we have used come from manuscripts that Engels left unpublished. It does not
seem possible to defer to this objection. Engels has expressed himself at length
on mathematics in his published works and there are no discrepancies between
his published works and unpublished manuscripts (more precisely, there are no
deeper discrepancies between the two parts than within the published part itself).
We may add that the Russian government published Engels' manuscripts a long
time ago and has used them just as much as the works published during his lifetime
to foster its official dogma.

The picture we have obtained consists of two parts, rather loosely joined. On the one hand, there is Engels' 'materialism', which reduces mathematics to physics, or rather to 'material observation', entirely ignores its if-then character and sees in it a kind of land surveying. On the other hand, there is the 'dialectic', which proclaims that mathematics breaks the rules of logic at every step and swarms with 'contradictions'. The 'materialism' is a very crude form of empiricism; the 'dialectic' is a degenerated offshoot of Hegel's philosophy. The only bond, it seems, that ties these two heterogenous parts together is a common ignoring of the real development of science.

Mathematics is undoubtedly the field in which Engels is at his weakest. His views on mathematics, however, are too deeply ingrained in his general conceptions to be dismissed lightly. They form a frame of reference that can never be forgotten in a general examination of his ideas.

In order to be complete the present study would require an examination of what Engels' conceptions have become when inherited by his epigones and commentators, as well as an examination of Marx' attitude toward mathematics.

The first task is too thankless to tempt us now. Suffice it
to say that the fate of Engels' writings has been determined by social considerations
rather than by a rational examination of their contents; only socio-political
events, not its intrinsic value, can explain why so mediocre a book as the *Anti-Dühring
*could become the philosophical Bible (if we may use these two words together)
of so many men. This is indeed an important social phenomenon (with which we
are not concerned here), but it does not in any way increase the intrinsic value
of the book.

The second task is full of interest and would require a special study; we simply give here a few conclusions. Marx left about 900 pages of mathematical manuscripts. A sizable part of these manuscripts were published in Moscow in 1968. Many pages are no more than abstracts from textbooks read by Marx. Some of his notes, however, consist of commentaries and deal with the definition of the derivative. Marx devised a method which he opposes to those of Newton, Leibniz, d'Alembert and Lagrange (he ignores Cauchy). His aim was, it seems, to decide whether a function 'reaches' its limit or not, a question long debated until the middle of the nineteenth century. As far as one can judge from the published manuscripts, Marx' method of obtaining the derivative involves no more than a change of notation, concealing the difficulty rather than solving it. By giving independent value to this procedure Marx only reveals that he has not yet fully grasped the notion of a limit; moreover, the method is applicable to polynomials only, not to all functions, and its use would make a general theory of the derivative impossible.

Marx' efforts are those of an alert student of the calculus, who tries to think a delicate point through by himself, but cannot yet undertake original creative work in mathematics because he lacks training and information. Still the mathematical level of these efforts is well above that of Engels' writings and, unlike Engels, Marx did not publish anything on mathematics.

Marx did, however, send some of his mathematical manuscripts on the definition of the derivative to Engels, who commented in a letter dated August 18, 1881:

I compliment you on your work. The matter is so perfectly clear that we cannot be amazed enough how the mathematicians insist with such stubbornness upon mystifying it. But that comes from the one-sided way of thinking of these gentlemen [Marx and Engels 1931, page 513].

How well these lines show their writer's cast of mind! Engels did not know anything of the development of mathematics during the fifty years (at least!) preceding the time he was writing. From all evidence, he would have been unable to even name the mathematicians of his time. Nevertheless, he does not hesitate to accuse them of incompetence. Marx' manuscript becomes 'a new foundation of the differential calculus' (Marx and Engels 1967, page 46) by a 'profound mathematician' (Engels 1935, page 10), while mathematicians, because of their ignorance of the dialectic, only muddle the problem.

This puts the finishing touch to our picture. Engels now stands as a man full of prejudices, unable to freely enter the competition of ideas. He would like to have his own 'dialectical' science aside from what he calls the 'ordinary metaphysical' science, that is, purely and simply science.

1.
At this place there is in the English translation of *Dialektik der Natur*
(Engels 1940, page 155) the following footnote: 'It is impossible to render
Engels' word "Induktionsesel" into English. A donkey in German idiom may mean
a fool, a hard worker, or both. It can thus imply praise and blame at the same
time. Probably, the implication is that Newton did great work with induction,
but was unduly afraid of hypotheses. The phrase might be freely rendered "Newton,
who staggered under a burden of inductions".' Of six persons with good knowledge
of colloquial German whom I have consulted, none has confirmed this version.

2.
After the publication of the *Anti-Dühring*, H.W. Fabian, a socialist
and a mathematician, wrote a very pertinent letter to Marx clarifying the point
(Engels 1935, page 719). Engels' only answer was a sneering remark in his preface
to next edition of the book (1935, page 10).

3. 'Abbilder', 'Spiegelbilder', 'Widerspiegelung'; Engels repeats these expressions time and again. See, for example, 1935, pages 24-26.

4. We leave aside the fact that the first proposition *is * no axiom at
all; it is false for infinite sets (with a certain sense of 'greater'). The
second statement expresses the transitivity of equality, one axiomatic property
among several. Curiously enough, the two 'axioms' cited by Engels are the two
*examples* of 'identical propositions' given by Kant in 1787, page 38.
Such ill-digested fragments abound in Engels' writings.

5. It would not be without interest to study Engels' ideas on heredity and his general attitude toward science in the light of the Lysenko affair.

6. Similar conclusions, although perhaps less complete, have been reached by other students of Engels' attitude toward mathematics; see Bataille and Queneau 1932, Hook 1937, Walter 1938 and 1948. In his 1934 Gustav Meyer says only a few words on the subject (pages 314-315), but they are very much to the point; see also 'Appendix B' in Wilson 1940.