From: glevy@PRATT.EDU
Date: Fri Apr 21 2006 - 08:05:10 EDT
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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