Ernst Kol'man and Sonia Yanovskaya (1931)

Hegel & Mathematics

Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Unter dem Banner des Marxismus, 1931.

The enormous interest shown in the study of Hegel by science in the Soviet Union is best justified in Lenin's philosophical legacy:

'Modern natural scientists (if they know how to seek, and if we learn to help them) will find in the Hegelian dialectics materialistically interpreted a series of answers to the philosophical problems which are being raised by the revolution in natural science and which make the intellectual admirers of bourgeois fashion "stumble" into reaction.'

If materialism wishes to be militant materialism, it must set itself such a task and work systematically at solving it, otherwise

'eminent natural scientists will as often as hitherto be helpless in making their philosophical deductions and generalisations. For natural science is progressing so fast and is undergoing such a profound revolutionary upheaval in all spheres that it cannot possibly dispense with philosophical deductions' [On the Significance of Militant Materialism]

Science and mathematics in the Soviet Union are uninterruptedly engaged in strengthening and extending their philosophical foundations with the help of the study of Hegel's dialectics from the materialist point of view, in order to continue their struggle against the pressure of bourgeois ideas and against the attempted restoration of the bourgeois world outlook as successfully and aggressively as it has so far.

What comes under consideration for the purposes of mathematics, besides various passages from the various works and from the Marx-Engels Correspondence, and particularly Anti-Dühring and The Dialectics of Nature and Lenin's philosophical works, is also Marx's previously unpublished manuscripts, of which the Marx-Engels Institute in Moscow possesses 865 closely-written quarto sheets in photocopy. Part of this work, mainly concerning the nature of differentiation and Taylor's Theorem, has already been deciphered.

How does the materialist dialectic assess the role of the Hegelian philosophy of mathematics Marxism-Leninism proceeds from the principle that:

'The mystification that the dialectic suffers in Hegel's hands in no way prevents him from being the first to present its general form of working in an all-embracing and conscious way. With him it stands on its head. One must turn it right side up again in order to discover the rational kernel within the mystical shell.' (Marx, Afterword to the Second Edition of Capital)

He therefore, of course, also considered Hegel's philosophy of mathematics from the point of view of a criticism that distinguishes, which knew how to separate the positive kernel of the material and its faithful translation and transformation from the negative shell of the mystically-distorted ideal. Thus we see the positive and the negative woven together in Hegel's philosophy of mathematics and we pose ourselves the task of freeing the materialist kernel from the idealist shell.

The attitude of the founders of Marxism to Hegel's mathematical views can be seen from the following quotation from Engels:

'I cannot pass over without a comment on old Hegel, who they say had no profound mathematical scientific education. Hegel knew so much about mathematics that none of his pupils were in a position to publish the numerous mathematical manuscripts among his papers. The only man to my knowledge to understand enough about mathematics and philosophy to be able to do that is Marx.' [Engels, Letter to A. Lange, March 29, 1865]

We dialectical materialists see the merit of Hegelian philosophy in the field of mathematics in the fact that Hegel:

  1. was the first to brilliantly guess the objective genesis of quantity as a result of the dialectic of quality;
  2. correctly determined the subject matter of mathematics and correspondingly also its role in the system of sciences and gave an essentially materialistic definition of mathematics which smashes apart the framework of the bourgeois world-outlook with its characteristic quantity fetishism (Kant and pan-mathematicism);
  3. recognised that the field of differential and integral calculus is no longer a merely quantitative field, but that it already contains qualitative moments and traits which are characteristic of the concrete concept (unity of internally contradictory moments); and that consequently
  4. any attempt to reduce infinitesimal calculus to elementary mathematics, to annihilate the qualitative leap between the two, must from the outset be regarded as ill-fated;
  5. mathematics, from its own resources, without the assistance of theoretical philosophical thought, is not in a position to justify the methods which it itself already uses;
  6. the origin of differential calculus was determined, not by the requirements of the self-development of mathematics, but its source and foundation are to be found in the requirements of practice (materialist kernel!);
  7. the method of differential calculus represents an analogue of certain natural processes and therefore cannot be grasped out of itself but only out of the essence of that field where this method finds its application.

The weaknesses, mistakes and errors of Hegel's view of mathematics, which follow with iron necessity from his idealistic system, rest, from the dialectical materialist point of view, on the fact that:

  1. Hegel believes that the method of differential calculus as a whole is a method alien to mathematics, so that within mathematics no transition can be created between elementary and higher mathematics; consequently however the concepts and methods of the latter can only be brought into mathematics in an external and arbitrary manner, through external reflection, and do not arise through dialectical development as a unity of the identity and difference of the new and the old;
  2. he thinks that such a transition is only conceivable outside of mathematics in his philosophical system, whereas by and large he is forced to carry the true dialectics of the development of mathematics over to his philosophical system;
  3. he often does this however in a distorting and mystifying way, and in doing so replaces the then still unknown real relations with ideal, fantastical relations and thus creates an apparent solution where he should have sharply posed an unsolved problem, and subjects himself to the task of proving and defending that in the mathematics of his day, which was often simply wrong;
  4. he considered the factual development of mathematics to be a reflection of the development of the logical categories, of these moments of the self-development of the idea, and denied the possibility of constructing a mathematics which would consciously apply the dialectical method and would therefore be able to discover the true dialectic of the development of its own concepts and methods and not simply take the qualitative and contradictory moments into itself through external reflection;
  5. correspondingly he is not only not in a position to pose the task of reconstructing mathematics through the method of dialectical logic, but he is forced to jog along behind the mathematics of his day despite his correct criticism of its basic concepts and methods;
  6. he prefers Lagrange's proof of infinitesimal calculus not because it uncovers the real relationships between the mathematics of the finite (algebra) and of the infinite (analysis) but because Lagrange brings the differential quotient into mathematics in a purely external and arbitrary way, whereby Hegel conforms to the usual shallow interpretation of Lagrange;
  7. he denies the possibility of a dialectical mathematics and in his efforts to diminish the significance of mathematics excessively, more than it deserves, he totally denies the qualitative (dialectical) moments in elementary mathematics (arithmetic). However, as their presence was obvious to a dialectician like Hegel, while he drove them out at one point (in the chapter on ‘Quantity’) he had to create them at another (‘Measure’).

Hegel's merit in correctly recognising the subject matter of mathematics deserves to stand high in our estimation, particularly in view of the fact that even today this question causes the greatest difficulties in the most varied idealistic and eclectic philosophical trends because they reflect material reality in a distorted way.

Thus the intuitionists (Weyl, Brouwer), following Kant, take the view that pure a priori intuition forms the subject matter of mathematics, while the logicists, who since Leibnitz take mathematics to be part of logic, see in axioms and theorems the laws of reason. The formalists, like Hilbert, deny the existence of a particular subject matter of mathematics at all, holding the latter to be a mere collection of rules that permit us to form various combinations and transformations. The mechanistic empiricists who classify mathematics as part of physics deny its specific nature, think that its subject matter is physical space and physical time. Others, like Mach, seek its subject matter in psychology, etc.

However, all these definitions lead to difficulties that none of these philosophical systems is able to overcome. As we know, the neo-Kantians (Bieberbach, Nelson) had to face not a few difficulties in order to reconcile pure a priori contemplation with non-Euclidian geometry. The logicists (Russell, Frege) were forced to take the view that mathematics was grammar without subject, object, verb and predicate, a grammar of the copula 'and', 'or', 'if', etc., in order to turn it thus into a gigantic tautology incapable of providing any new knowledge of the subject matter. The mechanistic empiricists were unable to classify multidimensional geometry in their system and were faced with the choice of recognising a mathematically possible geometry but excluding the rest from mathematics. The formalists, who have transformed mathematics into a sort of chess game with empty symbols, are not in a position to explain its role in technology, science and statistics. The conventionalists (Henri Poincare), who hold that mathematical concepts and operations are merely convenient, mentally economical conventions, thus avoid the question posed and are unable to make any statement about the development of these concepts.

Thus none of these philosophical schools, which all grasp one and only one side of reality, is in a position to understand the link between mathematics and practice and its laws of development. Hegel alone gave mathematics a definition such as grasped the essence of the matter, a definition which, quite independently of Hegel's views, is actually profoundly materialist.

According to Hegel mathematics is the science of quantity, i.e. of a determination of objects which does not describe them as such, in what makes them specifically different from other objects and from themselves at another stage in their development, but only from the side that is external and indifferent towards change.

‘Pure mathematics deals with the space forms and quantity relations of the real world - that is with material which is very real indeed. The fact that this material appears in an extremely abstract form can only superficially conceal its origin from the external world. But in order to make it possible to investigate these forms and relations in their pure state, it is necessary to separate them entirely from their content, to put the content aside as irrelevant.’ [Engels, Anti-Dühring, 1878, pp. 51-52]

This connection between mathematics and material reality reproduces the materialist interpretation of Hegel's definition of the subject matter of mathematics. The spatial relationships of our physical space correspond to the requirements of this definition, and spatial forms really are, according to Hegel, the subject matter of mathematics, even though they do not exhaust it, since any relationship that offers the possibility of various quantitative 'interpretations' can become the subject matter of mathematics. Thus for example the vortices dealt with by vector analysis can belong just as much to a fluid as to electrodynamics, which does not mean, however, that these mathematical vortices are a product of the idea, but that in themselves they reflect quantitative relations of real i.e. material reality.

Thus Hegel's definition grasps the actual essence of mathematics, provides the possibility of grasping its link with material reality and simultaneously shows the limits of mathematics, its place and role in the system of sciences which, as a whole and in their development, reflect objective (material) reality. From the standpoint of this definition the definitions quoted above can be not merely rejected a limine (from the threshold) but actually overcome. In each one of them moments of truth can be recognised, 'one of the features, sides, facets of knowledge' which, one-sidedly exaggerated and distended, develops 'into an absolute, divorced from matter, nature, apotheosised'. [Lenin, On the Question of Dialectics, Volume 38, Collected Works, p.363].

This can be done even though Hegel himself was not able fully to overcome the one-sidedness of these definitions. For in Hegel there are to be heard motifs which, often pretty eclectically jumbled, simply echo not only Leibnitz's logistics, but also Kant's construction from the elements of a priori contemplation, indeed even the conventionalist and formal denial of the objective correctness of mathematical statements. Thus he does in fact correctly describe the abstract, formal essence of the mathematical method, according to which “first definitions and axioms are set up, to which theorems are attached, whose proof consists merely in being reduced by the understanding to those unproven postulates”. [Hegel, Science of Logic]

But he himself one-sidedly exaggerates the moment of tautology in mathematics, closing his eyes to the evolution of this method which leads to the arbitrary and external character of the axioms being sublated - even though to this day the majority of mathematicians and philosophers of mathematics do not recognise this - and that in the development of mathematics the formal-logical moments of understanding are shouldered aside by the dialectical moments.

It is true that Hegel correctly notes the existence of the sensuous moments in mathematics, but he relies too much on Kant by reducing the content of mathematics like him to abstract sensuous intuition. For he agrees with Kant that mathematics 'does not have to do with concepts, but with abstract determinations of sensuous intuitions', wherein particularly 'geometry' has to do with the sensuous, or abstract intuition of space', which is true to the extent that the sensuous moment is particularly pronounced precisely in geometry, but which must not be made absolute even in relation to geometry. Moreover Hegel himself goes on to concede that even this science, which only deals with these abstract sensuous perceptions, 'nevertheless collides in its path, most remarkably, in the end with incommensurabilities and irrationalities where, if it wishes to proceed further in determining, it is driven beyond the principle of understanding'. (ibid.) Finally Hegel criticises, and rightly, the 'sleight of hand and charlatanry even of Newtonian proofs' which tried to present the laws of experience as the results of calculation. He is completely correct when he claims that by no means every single member of a mathematical formula, taken by itself, has to have a concrete significance and that the mathematical correctness of the result is no guarantee of the real sense (i.e. to which an existence would correspond) of the result of the calculation. But at the same time what this amounts to in Hegel is that in mathematical propositions in general he denies correctness as such in them themselves, that he considers mathematics, as do today's formalists, only from the aspect of its inner logical consistency, and not of its objective truth, i.e. only as a calculation, but not as a science which has its own subject of research.

Being the science of the abstract determination of quantity, mathematics can only portray one side of reality. Between it and physics there is already an essential difference, a node, a transition to the new quality. For physics already researches matter from the qualitative essential side. Its molecules, atoms and electrons are no longer indifferent relationships in which mutually differing things can emerge without changing their quality, but precisely molecules, atoms and electrons in the wholeness of their particularity, the specific way they arise and develop. Therefore physics cannot be reduced to mathematics; the role of mathematics in science is limited. This standpoint is diametrically opposed to that of Kant, according to which science is only worthy of the name to the extent that mathematics finds a place in it.

By coming out against the fetishisation of quantity, which after all is only a reflection of the abstract money-trading relations of the bourgeois order, Hegel in this case actually burst apart the framework of bourgeois philosophy. However, since he did not base himself on another class, but was and remained a philosopher of the bourgeoisie, he could only develop this, in its essence profoundly materialist standpoint, in an idealist way, and thus to unbridled hypertrophy. What was materialist in this standpoint of Hegel's is made particularly clear by the fact that it is precisely the notorious 'mathematicisation' of physics which has rendered the greatest service to idealism in philosophy and science. Not in vain did the natural philosopher Abel Rey, who despised materialism, write that 'the crisis in physics lies in the conquest of the realm of philosophy by the mathematical spirit' (Abel Rey, La Theorie physique chez les physiciens, Paris 1907, quoted in Lenin, Volume 14, p.309), a crisis in which 'matter disappears', only equations remain (ibid).

All the same, what had happened in science - the drawing together of the two sciences of physics and mathematics - was evaluated by Lenin as a significant success for science. This is in complete harmony with Hegel if we interpret him materialistically. Hegel it is true did not recognise the development of concepts in mathematics, since he did not count mathematics as part of philosophy, i.e. as a science dealing with 'concepts'.

“One could also conceive the idea of a philosophical mathematics knowing by Notions, what ordinary mathematics deduces from hypotheses according to the method of the Understanding. However, as mathematics is the science of finite determinations of magnitude which are supposed to remain fixed and valid in their finitude and not to pass beyond it, mathematics is essentially a science of the Understanding; and since it is able to be this in a perfect manner, it is better that it should maintain this superiority over other sciences of the kind, and not allow itself to become adulterated either by mixing itself with the Notion, which is of a quite different nature, or by empirical applications.” (Hegel, Philosophy of Nature, Miller trans., p.38)

But that does not mean that he completely overlooked this development. No, he merely transferred it from mathematics into his system of philosophy and here he demanded complete unity of development.

Between geometry and mechanics there must be a unity, everything must be linked by a chain of dialectical deduction, by the chain of development. Even the fact that our space has precisely three dimensions must find its explanation in the unity of development, but this cannot be achieved with the means of mathematics alone, but, as Hegel said, with the means of philosophy, as dialectical materialism maintains with the means of physics. Between physics and mathematics there is a unity of development and not of reduction, a unity of identity and difference. For not only the one science but the other too represents, as we maintain, real i.e. material reality at different levels of its complexity and development. The geometry of physical space and mechanics are two such fields, one standing directly above the other; between the principle of gravitation and the doctrine of the properties of material time-space there must therefore be a link, but at the same time a difference too. To discover this link we must develop further, 'physicise' it, if one may use the expression.

Einstein could not have developed his theory of relativity had not geometry progressed in the appropriate direction in which it filled itself with physical content. Riemann's differential geometry sublates' - using this term in Hegel's sense - Euclidian geometry by allowing the latter validity only as a moment, by subordinating and incorporating the geometry of 'rigid' unchanging space to and into the constant curvature of the geometry of a changeable 'fluid' space, which only remains Euclidian in its infinitely small parts, of a space where “either the reality on which the space is based forms a discrete multiplicity or the basis of the measure relations must be sought outside in forces operating on them to form them”, (ibid, p.284) where therefore bodies are no longer 'indifferent' in their mutual 'distances since the length of the path travelled depends on 'history' . It is not physics that is sublated and subsumed into mathematics, but mathematics that is developing and coming closer to physics by taking into itself more and more qualitative moments of measure. This development is therefore proceeding completely in the sense of the materialistically interpreted dialectical method of Hegel, even though it just as completely contradicts his system, which could not tolerate dialectics in 'conceptless' mathematics.

Thus the successes of the physical theory of relativity are no more to be linked to Hegel's idealist system than they are to be with the relativist philosophy, they came into being thanks to the spontaneous dialectics of the scientific researcher, which involuntarily reflects the true dialectic of nature. But the failures which Einstein's physical theory of relativity is suffering at the moment in its efforts to create an image of the world that adequately reflects reality and at the same time does justice to quantum relations, are based on an inability to grasp this reality as a unity of continuity and the discrete, on the obstinate desire to preserve it as the absolute continuum of ideal thought.

By removing dialectics from nature, from science, and transferring it to his philosophical system placed above nature, Hegel acts as a true idealist. For that very reason not only did he deny mathematics the ability to proceed in a consciously dialectical way but he also, despite his pronounced objectivism, falls into a purely subjective position in mathematics.

“To treat an equation of the powers of its variables as a relation of the functions developed by potentiation can, in the first place, be said to be just a matter of choice or possibility; . . . utility of such a transformation has to be indicated by some further purpose or use; and the sole reason for the transformation was its utility” (Hegel, Science of Logic, Miller trans., p.281)

- he wrote, in a style that we find again in Mach or Poincare. For the mathematically infinite, which emerges in mathematics in the form of the series, the transition of limit, fluxion, differential quotients, the infinitesimal, etc., is no longer something merely quantitative from his standpoint, but already contains a qualitative moment, so that here mathematics cannot avoid the concept, whereas the concept is supposed to be something alien to mathematics, something which is supposed to contradict all its laws, and thus mathematics can only take it in an 'arbitrarily lemmatic way' from a field alien to mathematics. Hegel correctly states that elementary mathematics would never have given birth to analysis out of itself, that it was driven to do so by the requirements of 'application', i.e. of practice, technique, science.

When Hegel writes: 'The appearance of arbitrariness presented by the differential calculus in its applications would be clarified simply by an awareness of the nature of the spheres in which its application is permissible and of the peculiar need for and condition of this application', (ibid., p.284) this materialist kernel is in completely the same sense as Engels's following claim concerning the material analogies of mathematical infinity:

'As soon, however, as the mathematicians withdraw into their impregnable fortress of abstraction, so-called pure mathematics, all these analogies are forgotten, it becomes something totally mysterious, and the manner in which operations are carried out with it in analysis appears as something absolutely incomprehensible, contradicting all experience and all reason.' [Engels, Dialectics of Nature, p.271]

But as a result of Hegel's idealist blinkers he does not notice, and in his time it was difficult to notice, how by this influence all the operations and concepts of mathematics came into motion and the whole mathematical edifice is renewed from the ground up. He correctly notes the failure of the attempts to assimilate the new concepts by the means of old ideas, but as a bourgeois philosopher who only intends to explain the world and not to change it, he does not at all pose himself the task of transforming mathematics dialectically.

'Until the end of the last century, indeed until 1830, natural scientists could manage pretty well with the old metaphysics, because real science did not go beyond mechanics - terrestrial and cosmic. Nevertheless confusion had already been introduced by higher mathematics, which regards the eternal truth of lower mathematics as a superseded point of view.' [ibid,. p.203]

So Engels claims, thus far agreeing with Hegel. But from here on the difference starts, because Engels goes on:

'Here the fixed categories dissolved, mathematics had arrived on a terrain where even such simple relations as that of mere abstract quantity, bad infinity, assumed a completely dialectical shape and forced mathematics, against its will and without knowing it, to become dialectical ' [ibid.]

According to Hegel these dialectical moments, which are alien to the elementary mathematics of constant magnitudes, cannot be adopted by mathematics at all. All the attempts by mathematics to assimilate them are in vain, for since mathematics is not a science of 'concept', therefore naturally no dialectical development, no movement of its concepts and operations on its own ground is possible, and the only possibility that remains open to it is to 'agree upon a 'convention' arbitrarily, according to Lagrange to designate 'derivatives of given primary function' as the coefficients of a particular member of the development of Taylor's series of that function. At best what can be shown in this is the convenience and suitability of precisely that and no other 'convention'.

The great dialectician correctly criticises all the attempts undertaken in his day to prove analysis, but in doing so he does not draw the expected conclusion that these attempts failed because they did not develop analysis dialectically but tried to reduce it to elementary mathematics. He concludes rather that this is impossible in the field of mathematics, and that it is only possible in the interior of philosophy and in his system of categories developing out of one another. While driving dialectical development out of mathematics in this way and transferring it to his system of pure categories of logic, he often subjects it to quite abstruse, sophistic and fantastic mystification. As an example of this one only needs to read how intensive quantity, after uniting with its opposite, extensive quantity, goes over to an infinite process, and more of the like. Hegel's artificial, mystical and mystifying transitions confirm in this field too that idealist dialectics, which aims to develop concepts out of themselves does not reflect real relations and transitions, the movement and development of material reality, becomes fruitless because of its idealist moment; that there can be no scientific dialectic other than the materialist dialectic.

However, by annihilating the inner dialectic of concept in mathematics Hegel deprives himself of the opportunity of revolutionising mathematics, at least in the interior of his philosophical system, and is forced merely to transfer passively and to 'prove', instead of actively working and transforming, and at the best to propose a change of name, like for example 'development function' instead of 'derivative'. When Hegel claims that in the interior of his system of logical categories he has not only proved the possibility but has also given the true substantiation of that same mathematical infinite in all its varieties on which all previous attempts to substantiate analysis had come to grief, in fact he himself is labouring under the same mental images against which he polemicises so sharply. Thus for example he is right when he condemns as unscientific and anti-mathematical the method of neglecting infinitesimals of a higher order on the basis of their quantitative insignificance and when he declares the same method to be permissible on the basis of the qualitative meaning of these magnitudes. Since the differential is a quantitative-qualitative relation, in the development

(x + dx)n - xn = nxn -1dx + nn-l/1.2 . xn-2dx2 + . . .

the form of sums appears as something external and unessential, from which therefore abstraction must be made. 'Since what is involved is not a sum, but a relation, the differential is completely given by the first term ,' he writes (op. cit, p.265), and thus rescues himself with the same dodges and bolt-holes of which he completely accuses the creators of infinitesimal calculus, whom in fact he follows, at great pains to let in at the window what he has just thrown outset the door.

Precisely because Hegel, starting from his idealist standpoint, did not pose the task and could not pose it of reconstructing mathematics by means of dialectical logic, but only tried to 'substantiate' it in the interior of his philosophical system as it stands, he never achieved even this task, despite a whole number of the most valuable comments, and had as good as no direct influence at all on the further development of mathematics although the latter, as we have already shown, was spontaneously proceeding precisely along a dialectical path.

What is much more responsible for the fact that Hegel's dialectic exerted no influence on the development of science and mathematics is the bourgeois narrowness that treated him like 'a dead dog'. This led to the situation where all that has remained alive from Hegel's works is what Marx and Engels as the ideologists of the proletariat have stood from its head on to its feet from his teachings and have placed at the service of the proletarian revolution.

By overcoming the idealist dialectic in a materialist way, Marx, Engels and Lenin were enabled, in contrast to Hegel, to bequeath us truly scientific theoretical statements, i.e. appropriate to material reality, to practice, in the field of mathematics too, which serve us as guidelines for research, scientific prediction and creation. The nodal points here are formed by the Marxist-Leninist conception of the sources and powers of development of mathematics, of its essence, the interconnection and significance of its parts, of what is dialectical in mathematics itself and of the role that mathematics has to play in relation to other sciences

“But it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is, to perform the first arithmetical operation, are anything but a free creation of the mind. Counting requires not only objects that can be counted, but also the ability to exclude any properties of the objects considered except their number - and this ability is the product of a long historical evolution based on experience. Like the idea of number, so the idea of figure is borrowed exclusively from the external world, and does not arise in the mind out of pure thought. There must have been things which had shape and whose shapes were compared before anyone could arrive at the idea of figure . . . Like all other sciences, mathematics arose out of the needs of men: from the measurement of land and the content of vessels, from the computation of time and from mechanics. But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, became divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform. That is how things happened in society and in the state, and in this way, and not otherwise, pure mathematics was subsequently applied to the world, although it borrowed from this same world and represents only one part of its forms of interconnection - and it is only just because of this that it can be applied at all.” [Engels, Anti-Dühring. pp. 51-52)

And further on:

'The mystery which even today surrounds the magnitudes employed in the infinitesimal calculus, the differentials and infinities of various degree, is the best proof that it is still imagined that what we are dealing with here are pure "free creations and imaginations" of the human mind, to which there is nothing corresponding in the objective world. Yet the contrary is the case. Nature offers prototypes for all these imaginary magnitudes.' (Engels, Anti-Dühring, p.436)

This conception naturally has nothing in common with that of empiricists such as J. S. Mill, since unlike theirs it does not limit cognition to induction, but in contrast to the 'pan-inductionists' that Engels laughs at considers the logical as the historical worked over.

Thus mathematical concepts and conformities to law are considered not as absolute, unchangeable, eternal truths, but as parts of the ideological superstructure of human society tied to the latter's fate. It thus goes without saying that the main law of social development, the law of class struggle, cannot remain without influence on mathematics.

'There is a well-known saying that if geometrical axioms affected human interests attempts would certainly be made to refute them. Theories of the natural sciences which conflict with the old prejudices of theology provoked, and still provoke, the most rabid opposition.'

This standpoint, which thus has nothing in common with the claim by Kautsky and Cunow that mathematics and the natural sciences must be counted completely among the forces of production, which is the same as denying the class struggle within them, rejects the division of sciences into exact - mathematics and the natural sciences - and not exact - the social sciences.

The class standpoint in mathematics must not, however, be interpreted in such a way that all previous mathematics is rejected as a whole and that in its place a mathematics constructed out of completely new elements must be set up according to totally new principles. We take the position that the development of mathematics is determined by the developing productive forces (whereby mathematics itself has a reciprocal effect on the productive forces) and consequently reflects material reality. However, the productive forces exert their effect on mathematics by means of the connecting link of the production relations, which in class society are class relations and stamp the distorting class impress on mathematics. Thus mathematics displays a dual nature.

“Philosophical idealism is only nonsense from the standpoint of crude, simple, metaphysical materialism. From the standpoint of dialectical materialism, on the other hand, philosophical idealism is a one-sided, exaggerated, überschwengliches (Dietzgen) development (inflation, distension) of one of the features, aspects, facets of knowledge into an absolute, divorced from matter, from nature, apotheosised . . . Human knowledge is not (or does not follow) a straight line, but a curve which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line, which then (if one does not see the wood for the trees) leads into the quagmire, into clerical obscurantism (where it is anchored by the class interests of the ruling classes). Rectilinearity and one-sidedness, woodenness and petrification, subjectivism and subjective blindness - voila the epistemological roots of idealism. And clerical obscurantism (philosophical idealism), of course has epistemological roots, it is not groundless; it is a sterile flower undoubtedly, but a sterile flower that grows on the living tree of living, fertile, genuine, powerful, omnipotent, objective, absolute human knowledge.” [Lenin, On the Question of Dialectics, Collected Works, Vol. 38, p.363]

All the less can bourgeois mathematics be simply rejected, but on the contrary it must be subjected to a reconstruction, since it represents the material world, albeit one-sidedly and distortedly, nevertheless objectively.

But if mathematics owes its origins to practice, if it reflects real relations and conditions derived from material reality (albeit in a completely abstract and distorted form), therefore it must be dialectical. For 'dialectics, so-called objective dialectics, prevails throughout nature' [Engels, Dialectics of Nature, p.211], and 'the dialectics in our head is only a reflection of real development which takes place in the realm of nature and of human society and which follows the dialectical forms' [Letter to Konrad Schmidt, November 1, 1891]. 'This mystical in Hegel himself, because the categories appear as pre-existing and the dialectics of the real world as their mere reflection' [Dialectics of Nature, p.203]. And actually as we have already said, Engels held that higher mathematics was dialectical since the ntroduction of variables by Descartes brought into them at the same time movement and therefore, also dialectics. Hegel correctly noted that new qualitative and - dialectically internally contradictory moments thus penetrated into mathematics. But he overlooked what Engels emphasised, that is to say that mathematics itself was thus forced, although unconsciously and against its will, to become dialectical and that therefore the dialectic of the development of its basic concepts and methods must-be sought within mathematics itself.

Nevertheless, elementary mathematics, just like formal logic, is not nonsense, it must reflect something in reality and therefore it must contain certain elements of dialectics. Engels too can actually see it, in contrast to Hegel.

'Number is the purest quantitative determination that we know. But it is chock full of qualitative differences . . . 16 is not merely the sum of 16 ones, it is also the square of four, the fourth power of two . . . Hence what Hegel says [Quantity, p.237] on the absence of thought in arithmetic is incorrect.' [ibid., pp 258-259]

Even in elementary algebra and arithmetic he sees a 'transformation of one form into the opposite' which is 'no idle trifling' but 'one of the most powerful levers of mathematical science without which today hardly any of the more difficult calculations are carried out' [ibid., p.258]

Marx however saw, not only in agreement with Hegel, both the impossibility of all attempts to provide a formal-logical substantiation of analysis and also the - childishness of trying to make it rest on sensuous intuition, on the graphic, etc. He not only fought for the dialectic of mathematics, particularly of analysis, but more than that he undertook an independent attempt to build up a dialectical foundation based on the unity of the historical and the logical. In doing so Marx poses himself the task, as we have already mentioned in passing, of not reducing analysis to arithmetic, as the logicists, starting with Weierstrass, later tried to do, which, despite all their achievements in deepening the way in which mathematical problems are posed, led to the well-known paradoxes of set theory which destroyed the whole structure, not only of mathematical but also logical, which had been specially erected for that purpose. Marx tries to show how the essentially new differential and integral calculus grows out of elementary mathematics itself and out of its own ground, appearing as 'a specific type of calculation which already operates independently on its own ground', so that 'the algebraic method therefore inverts itself into its exact opposite, the differential method', and in this way as a leap that 'flies in the face of all the laws of algebra'. 'This leap from ordinary algebra, and besides by means of ordinary algebra, into the algebra of variables ... is prima facie in contradiction to all the laws of conventional algebra.

Just like Hegel, Marx is closest to Lagrange in his proof of analysis. But his conception of Lagrange is fundamentally different from Hegel's conception. Hegel conceives Lagrange, as we have already seen, according to the usual shallow interpretation, so that Lagrange appears as a typical formalist and conventionalist introducing the fundamental concepts of analysis into mathematics in a purely external and arbitrary manner. What Marx admires about him, on the contrary, is the exact opposite; the fact that Lagrange uncovers the connection between analysis and algebra and that he shows how analysis grows out of algebra. 'The real and therefore the simplest connections between the new and the old', Marx writes 'are always discovered as soon as the new takes on a rounded-out form, and one can say that differential calculus obtained this relation through the theorems of Taylor and MacLaurin. It thus fell to Lagrange to be the first to reduce differential calculus to a strictly algebraic basis.' But at the same time Lagrange is criticised by Marx for overlooking the dialectical character of this development and staying too long on the ground of algebra and disparaging the conformity to law and method of analysis itself. For that reason 'he can only be used as a starting point in that respect'. Thus Marx, the true dialectician, fights on two fronts here too: against not only the purely analytical reduction of the new to the old, which was so characteristic of the mechanical methodology of the 18th century, but also against the purely synthetic introduction of the new from outside, which is so typical of present-day intuitionists also, which presents the principle of complete mathematical induction as that which is new, coming from outside, from intuition and thus obliterates the transition between logic and mathematics. Here too Marx fights for dialectical unity, for the unity of analysis and synthesis.

From the dialectical materialist conception of mathematics as a depiction, although extremely abstract, of the laws of motion of material reality, it follows that dialectical materialism has a much higher estimation of the role of mathematics than Hegel did. Engels particularly emphasises that 'a knowledge of mathematics and natural science is necessary for a conception of nature which is dialectical and at the same time materialist', [Anti-Dühring] although he does not overlook the difficulties of applying it to the various branches of knowledge and particularly emphasises that 'the differential calculus for the first time makes it possible for natural science to represent mathematically processes and not only states'. [Dialectics of Nature]

The increasing difficulties offered to the mathematics of complicated forms of motion, piling up in an ascending series in leaps from mechanics to physics, from physics to chemistry, from there to biology and onwards to the social sciences, do not, in the dialectical materialist conception, entirely block its path, but allow it the prospect of even 'determining mathematically the main laws of capitalist economic crisis' [Marx, Letter to Engels, May 31, 1873].

Dialectical materialism considers the dialectic of concepts as only the conscious reflection of the dialectical movement of the real world, and holds this interconnection to be valid, the determination of the ideal by the material, of theory by practice as the leader in the final analysis. It therefore follows that the standpoint of dialectical materialism on the further development of science in general and also of mathematics is the direct opposite of the standpoint of Hegel. Whereas Hegel merely tries to substantiate what already exists, it is a matter here of a transformation, the conscious change, the reconstruction of science on the basis of practice. This attitude, which sharply distinguishes Marxism-Leninism from Hegel's philosophy and all other idealist and eclectic world-outlooks, enables it to see new paths of development in the territory of the individual sciences and to protect science from stagnation and decay.

Present-day science, the natural science and mathematics of the capitalist countries, is, just like the whole capitalist economic and socio-political system, shaken by a crisis unparalleled in both its extent and its profundity. The crisis of science, which itself serves as the best testimony against the widespread but completely unfounded belief that the natural sciences, like philosophy, are supposedly independent of politics, shakes above all at the methodological roots. The panic and the lack of perspective gripping the minds of the ruling class in the social field is reflected in science, in the flight of the majority back to mysticism, while 'a portion of the bourgeois ideologists who have raise themselves up to the level of comprehending theoretically the historical movement as a whole . . . goes over to the proletariat' (Marx and Engels, Communist Manifesto), strives to grasp its world outlook and methodology, dialectical materialism, and to impose it in science, and naturally feels itself drawn to the science of the victorious proletarian revolution. The present-day crisis of science is, however, destroying not only the philosophical justification of science, but the skeleton of science itself. Not only does it deprive it of material means and labour power, but it drives its thematics into the blind alley of perspectivelessness, bringing ever closer the peril that the apparatus of scientific theory itself will be blunted and will prove unable to solve the problems of practice.

Thus Bertroux [P. Bertroux, L'Ideal Scientifique des Mathematiciens, 1920] for example shows the ways in which the mathematician chooses his themes nowadays, and comes to the disconsolate conclusion that the overwhelming majority of the new mathematical works consists in small improvements and enlargements to and analogies of older works, that the method of mathematical research that even Leibnitz complained of, which leads to a flood of essays and to 'disgust with science', has gained and is gaining ground, but that no other paths can be recommended to mathematicians, but that they should continue to rely on 'the general tendencies of science of their age'. The origin lies in the separation in principle of theory from practice peculiar to idealist philosophy, in the stigma of planlessness borne by the entire capitalist system as a whole. Only a philosophy which adopts the goal of adequately depicting the movement of material reality can serve science as a reliable beacon to preserve it from the deadly separation from practice, from the 'evergreen tree of life'. Only the principle of planning, whose introduction is incompatible with the principle of the private ownership of the means of production, with the dictatorship of the minority over the majority, can save science from withering in empty abstractions and, by unleashing the powers of scientific talent slumbering in the popular masses, bring it to a new and unimagined bloom.

Science in the Soviet Union, and mathematics as part of it, is strong for this very reason that it possesses the dialectics of Hegel, materialistically overcome and freed from idealist distortions, and the principles of socialist planning, which for their part translate into reality the doctrines of dialectical materialism, as a guideline, and new, numerically growing mass cadres of the proletarian student body, bringing forth new scientific powers out of themselves, as bearers. The carrying out of the Five Year Plan, the electrification of the Soviet Union, the construction of new railways, the setting up of giant metallurgical works, of coal mines, etc., the industrialisation of collective agriculture, the construction of socialist towns, the polytechnicalisation of the schools and the liquidation of elementary and technical illiteracy, all this poses mathematics a great number of questions which will be successfully solved in a planned way, with the collaboration of all branches, in collective work and guided by the sole scientific methodology of the materialist dialectic, and will be able to have a fruitful effect on the development of mathematical theory.

Thus the philosophy of Hegel is materialised in both meanings of the word in the Soviet Union: as to its content, and as a mess act through the proletarian dictatorship. As such, however, it is the guarantee that what is immortal even in Hegel's mathematical thoughts, from the private property of a privileged caste of academic, protected by a mystic veil, will become the common property of millions of toilers.