**From:** Paul Bullock (*paulbullock@EBMS-LTD.CO.UK*)

**Date:** Sat Apr 22 2006 - 18:21:11 EDT

**Next message:**glevy@PRATT.EDU: "Re: [OPE-L] Dumenil and Levy on Unproductive Labor"**Previous message:**Jurriaan Bendien: "[OPE-L] Dumenil and Levy on Unproductive Labor"**In reply to:**glevy@PRATT.EDU: "[OPE-L] Friedrich Engels, Karl Marx and Mathematics"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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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