From: Paul Bullock (paulbullock@EBMS-LTD.CO.UK)
Date: Sat Apr 22 2006 - 18:21:11 EDT
Thanks Gerry,
I 'm always happy to read criticism such as this. No doubt Engels was weak
in maths..but clearly Heijenhoort has his views on mathematical methodology
and a philosphy which is quite speculative.... As usual I am indebted to
your assiduity.
Paul B.
----- Original Message -----
From: <glevy@PRATT.EDU>
To: <OPE-L@SUS.CSUCHICO.EDU>
Sent: Friday, April 21, 2006 1:05 PM
Subject: [OPE-L] Friedrich Engels, Karl Marx and Mathematics
> Recalling the speech given at Marx's grave by Engels, Paul B asked
> some time ago about Marx's "original" contributions to mathematics.
> The following article by Jean van Heijenoort deals with that issue
> and also calls into question FE's knowledge of math. [*Warning to
> Paul B: you are not going to like this article!*] van Heijenoort was
> Trotsky's secretary from 1932 to 1939. He received his PhD in math
> from NYU in 1949 (the year after the following article was written)
> and went on to teach in the Mathematics Dept. at NYU until 1965,
> then taught in the Department of Philosophy and the History of Ideas
> at Brandeis University. He died in 1986. The following article was
> published on the Net by the Marxists Internet Archive (MIA).
>
> Did van Heijenoort get it right about the knowledge of E & M about
> math?
>
> In solidarity, Jerry
>
> Jean van Heijenoort-Friedrich Engels And Mathematics
> Jean van Heijenoort
> Friedrich Engels And Mathematics
> (1948)
> 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.
>
> Engels' mathematical knowledge
>
> 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 Francour'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]. *
>
>
> * 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.
>
> 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]. *
>
> 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
>
>
> * 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).
>
>
> 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, A3/T2 is the constant. Let us write it A.A.2/T2
> and, following Newton, let us call A/T2 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.
>
> The nature of mathematics
>
> 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' * of material things, mathematical
> concepts in particular are nothing but 'imprints of reality' (1935, page
> 608).
>
>
>
> * 'Abbilder', 'Spiegelbilder', 'Widerspiegelung'; Engels repeats these
> expressions time and again. See, for example, 1935, pages 24-26.
>
>
> 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
>
>
> that two times two makes four, that the three angles of a triangle are
> equal to two right angles, that Paris is in France, that a man left
> without food dies of hunger [1935, page 91].
>
> 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. * 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.
>
>
> * 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.
>
>
> 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. *
>
> Logic and mathematics
>
> Engels divides mathematics into two parts, 'elementary mathematics, the
> mathematics of constant magnitudes', and 'higher mathematics', 'the
> mathematics of variables, whose most important part is the infinitesimal
>
>
> * 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.
>
>
> calculus'. The two realms use different methods of thought: 'elementary
> mathematics [...] moves within the confines of formal logic, at least on
> the whole', while 'higher mathematics' is 'in essence nothing else but the
> application of dialectic to mathematical relations' (1935, page 138). The
> dichotomy of mathematics parallels the division of thought into
> 'metaphysical' and 'dialectical':
>
> 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 A1/2 = [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.
>
> Conclusion
>
> 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. *
> 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
>
>
> * 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.
>
>
> 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.
>
>
> ----------------------------------------------------------------------
> SOURCE: Van Heijenoort, Jean. "Friedrich Engels and Mathematics", in
> Selected Essays (Napoli: Bibliopolis, 1985), pp. 123-151.
>
> Note: The list of references for the whole book as well as this article,
> not reproduced here, is given on pp.153-166.
>
>
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