Marx and Mathematics

The story of man and his thought, of the world and truth, may begin anywhere and anytime. In any way, even. The mathematical way for instance. Even today, in the remote caves we find here and there in the world we can see on the walls mathematical inscriptions, in all probability made by pre-historic man. A few vertical lines on the wall crossed over by horizontal line, similar to some kind of addition. In much the same way as card players keep track of their score. In as far back as Ancient Greece, people used to draw, perhaps not always on papyrus or parchment but at least on sand whole triangles, even circles and their sections, correlations. They were entirely expressed through words, for instance: the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. The idea of recording these same sentences in abbreviated form represented in itself a step forward. The ar. s eq. 1/2 prod. base and hei. — meant, of course, that the area of a triangle was equal to half of the product of the base and height. It was only with Viιte and finally with Descartes that the symbol ah/2 was introduced which was neither at all a new number, nor an individual relationship but merely a general abbreviation for something that had already been known. This mode of abbreviation was, nevertheless, a huge condition, whole equations could be written in a concise and succinct manner (c = a + b is the simplest example of this) and operate with them further — unless this were so, differential and integral calculus would be impossible. The whole notion of the difference of the same dependent or independent variables, for two different values that are independently variable if it is itself dependent, for example, it was possible to mark simply with Δ — delta. This is no longer even a general relationship, no defined relationship, let alone a special law, this is simply an agreed upon designation. If the interval of observation of the independent value tends towards zero under such and such circumstances and so on — the whole notion about all the circumstances — and the very difference of the dependent value naturally tends towards zero, that is the increment is infinitely small: d, once again on the basis of an agreement. For instance, Δx or db, the increment of a value x, or an infinitely small increment of value b, reads: delta x or differential b etc. There is no mention, not even in the most general terms of calculus, let alone of some numerical value — this is but a language, a mathematical language instead of this or that defined, not only one-national but long since quite inappropriate one.

Almost the whole of the 20th Century, a.d. when man first set foot on the Moon, and man himself floated in space circling around Earth, this century was marked, on the other hand, by an equally interesting not so much phenomenon but condition for every phenomenon, a general atmosphere, so to speak. As if every story about man, his thought, suffering, struggle and history just had to start with 1918, 1934 or 1941, depending on the country discussed, be it R, C or any other. Every man feels his own pain the most, naturally enough, and every human wound, sacrifice and victory deserves the greatest degree of respect, particularly when a whole nation bleeds in conflict and in war, yet it is that strange atmosphere which seemed, not only rarely and also not only in one country, to have determined the mode of the story. All contemporary human knowledge and skill would appear to have at least in embryo, at least in terms of idea and proposal, stemmed from Marx and the whole story to have been confined to Marxism. And the less one knew of what Marxism genuinely was and what its boundaries were, at least the basic theoretical, rather than ideological, features, atmosphere were to be felt with greater conviction, transcending all the frontiers which that great thinker of the 19th century set - a lawyer by education, a philosopher by doctoral thesis, economist by his works, a writer by public impact of the Manifesto of the Communist Party, had been able to imagine. Thus for instance an advertisement appeared in the newspapers on March 1. 1982: here and here, and just not anywhere but at the People's University, at this and that hour, a public debate would be organized entitled: The Topical Interest of Marx's Mathematical Manuscripts. There is no doubt about it, Marx was a man of tremendous work ability and energy and had a wide range of occupations. Such a man, even when learning mathematics in his leisure hours or out of curiosity, would study the basic notions of differential calculus, in terms of definition and table cases, in detail, making his own notes here and there. To bring to light and comment these pupil-like notes, however, with pretensions to boot, in a situation when not even Marx in his three volumes of Capital, more voluminous than the Bible, for instance, by 25 per cent or of War and Peace, by 40 per cent, applied no more than percent calculations, would indeed be a sin of sorts towards Marx and an act of violence against his thought. The whole exercise, circus, that is, lay not in the endeavor to make the manuscripts appear as a historical curiosity — in much the same way as it would be interesting to peer into the mathematical manuscripts of a Tolstoy, for example, or Napoleon — but rather to endow them with a topical meaning.

The big hall, rows and rows of seats, near the rostrum and at the rostrum microphones. A number of extra tables were set up for the occasion, covered with green baize table covers. On them were vases with flowers. Behind them red velvet curtain drawn to the side. The blackboard cleanly erased and gleaming. New pieces of chalk. The maths lesson could begin. And the distinguished scholars, not ordinary scholars but those from this and that university department and institute, brimming with erudition and distinction, all economists to the last, speaking enthusiastically about Marx's contribution to mathematics. In this and that letter to Engels, Marx mentioned mathematics when pointing to some difficulties or rather possibilities. Marx added and divided, while criticizing Hegel's dialectics and the maximum transition of mathematicians to the differential, in the attempt to be as accurate as possible in his calculation, Mark wrote in a letter to Engels that one would have to take the greatest possible number of samples. And we know full well today that the integral is the infinite sum of infinitely small, and hence Marx was on the right path and so on. For, today, mathematical statistics and linear programming have indeed become integral parts of the science of economy and so on. And then the question: has anyone anything to ask? And then a rather corpulent man somewhere from the ninth or tenth row rises with difficulty and begins also with difficulty: He is not clear about something, with all due consideration, he had done some reading himself and as far as he could gather in the whole of Marx's theory the basic notion was value, that is surplus value. Everything revolves around that since from surplus value derives profit, exploitation and so on. So it is not clear to him how this surplus value is calculated. The question was whether Marx in fact calculated surplus value and according to which method, and whether this was at all possible? To this short and straightforward question the reply had to be equally short and straightforward. And it was indeed a short one. Yes, began Doctor So and So, with his formal tie, formal suit and equally formal and solemn attitude — Marx expressed this accurately: v plus delta v — was his word for word answer. Surplus value is therefore delta v — he concluded without batting an eyelash. He did not even use his piece of chalk, this magic formula remained to float solemnly in the air. Fatso's jaw dropped. And yet the answer was clear, indeed everything was clear. Here any further discussion ended. The professors were no longer in the mood to talk. The maths lesson was over. Applause and — good night!

Under such circumstances, however, in an atmosphere that always emerges anew as of itself and is created from nothing real thought has difficulty in articulating itself, any attempt at critical hinting is akin to an attempt at stirring up a wind with a hat. Marxism as a totality of human thought and desire, all the way up to every-day politics and perhaps especially then, has nevertheless had the pretension of grounding itself in the most narrow manner in philosophy, even in the ontological sense. Eventhough this ambition appeared somewhat later and not as a major preoccupation, as seemingly a still greater need to put a full stop on that indeed purely philosophical set of problems, always discussing in the same way and without a true cognizance of the perpetual movement of matter which is eternal. This in fact clearly descriptive axioms about matter which are largely found only in Engels' manuscripts should be the substance and correct result of all human philosophical thought about nature, and when it is stated that matter is infinite everything is said. Lenin added to this the definition according to which matter is an objective reality and thus the common sense system of philosophy was rounded off, that which is not merely a criticism of Hegel's or any other philosophy but that which should in itself be a philosophy. This so-called dialectical materialism, in short and widely referred to as a dia-mat constituted the philosophical-ontological basis underlying Marxism which has played an important positive role in politics and which endeavors to play it, that part, in every-day ideology. Hence it is all the more difficult for independent and true i.e. not exactly every-day thought, that which attempts to transcend every human and worldly division and boundary at least in the philosophical sense and therefore finds it increasingly hard to find its way. If matter is naturally both eternal and, of course, incapable of being created and infinite, why is it thus and what is it in fact, what does to be infinite actually mean? How is it that it exists at all and what does of its very self imply? This entire area, the true realm of philosophy about how to understand or at least how to experience those axioms concerning that (really) God-Matter, in fact the entire domain of philosophy, dialectical materialism leaves empty, whereas every-day Marxist ideology even negates it. The author of these lines felt the irresistible urge and the most intimate need to penetrate this realm and to solve questions relating to matter. Seeing as how authentic works of philosophy, by philosophers who as such have withstood the trials of history, had been mainly unknown to him and seeing as how dialectical materialism would have anyway been the ultimate result of that history, a result with which, incidentally, the author was not satisfied, he embarked upon an independent adventure. Thus, almost ten years ago a novel was born, a philosophical novel admittedly, entitled Essay on God or About the relativity and symmetry relating to Grandpa John's Death. While at school, the author was not interested in philosophy, the boring story about man's stupidity and the senselessness of idealism of times past, in a situation when ultimately all was solved well, not terribly interesting but nevertheless true. That is why he conceived his work as a novel: an intimate settling of accounts of that which in the minds of the wise had long since been clear and where they had already put a full stop. At long last the writer himself was satisfied. Those six or seven axioms about matter he managed to reduce to three: about relativity, about symmetry and about their synthesis, a synthesis which however leaves certain things incomplete and poses new questions, which (is it really?) indicates (is it possible, really?) that there are no basic axioms. And yet, the writer was satisfied: he had experienced foreboding as a solution he was able to believe in like others believe in God, a foreboding as a replay to a question about How was it that the World was in no way created and for no reason and that it further means nothing but nevertheless does exist? And then he noted with horror that nowhere in the Essay had he defined matter. The fact that he had not felt the need to define it he considered to be an unpardonable shortcoming. Later on, he opted for the definition that matter was the mainstay of possibilities, a definition that was not after all so bad to the extent that the definition of matter had at all any sense, a definition that had, in a slightly different way, been provided Bloch, for instance. It was only later that the writer of that Essay and of the present Philosophy came to realize that no definition of matter could have any sense, that this was not at all necessary for genuine philosophy, in much the same way as the definition of idea, for that matter. Still, this does not imply that the writer would, in a renewed edition of the Essay leave out the definition of matter. No! Although some improvements might really be made, some parts and indeed a softening of style perhaps — since the author had the occasion in the meantime to acquire philosophical knowledge, had read all sorts of things from the original philosophical treasure which had accumulated throughout the years — he would nevertheless leave the definition of matter. One novel, one work of art, one philosophy, as a matter of fact, is a unique and singular act much like the moment of life which in some way goes on in the most beautiful, most perfect, most comprehensive, most authentic manner. The written character is but a written sign of all that. When that sign is finally put down, the experience comes to an end. Subsequent experience, even of the same work, is but a new artistic, a new philosophical work. Hence subsequent corrections make only sense if they are identified with the earlier work, if it helps the latter to be more articulate. The writer would therefore leave the definition of matter but would only in harmony with the Essay, alter it for instance with the later definition of vacuum. Thus: Matter is the only, admittedly the sole way in which possibilities may exist. Today, taken by and large, the author does not feel it an omission that here in PHILOSOPHY AND BELIEF he had failed to define matter. On the contrary, he considers this a quality of the work and a quality of philosophy. Matter has however nevertheless been defined here in the epilogue. Not only for artistic and creative reasons in view of the Essay but also given the practical implications. If indeed, one insists at every cost on idealism and materialism, on division, that is, this one and precisely this one, then the writer regards himself a materialist.

Yet, under the circumstances described, in the ascribed atmosphere it is not always enough to be a simple materialist, it is not rare that one must at all cost be a Marxist, so as in this way to safeguard the last hope that the manuscript will be published. The writer does not consider himself in the least an "anarchy-liberal" or "counter-revolutionary", a "revisionist" or "enemy of socialism", but indeed a Marxist. If it is true what Lenin wrote that Marxist teachings were powerful because they were authentic — and even if this were to not only apply to the frameworks and be a criticism of the capitalism of the time but rather generally and for all times — then this work, Philosophy and Belief, is Marxist because it is authentic. Admittedly, truth does not only have one face, and one often does not know which is the real one, but which one is real and which one is the only one, is decided upon by others. The writer assures that the will value and respect any decision and any assessment about it. Every one that relates to the work and possibly a different perception of the truth of that work. Everyone, except that which encroaches upon the intentions of the writer himself, although the writer dares dispute the validity of Lenin's definition of matter, its philosophical and not its political and educational validity, although he dared question Marx's eleventh thesis on Feuerbach while he took the second as a motto, although lie dared question the soundness of Engels' approach to philosophy, the writer's intentions were nevertheless and precisely because of this exclusively intellectual, out rightly honest and extremely conscientiously reflected in his work. Although he wrote "neither capitalism nor socialism", the writer is not against socialism. If socialism implies freedom and general progress, how could he be against freedom and progress? The writer merely pointed out that provided there was tolerance, man would not need, not be bothered by (such) a capitalism or (such) a socialism. The writer only hopes, if at all he dares express that hope, that the people who conduct in a responsible manner those responsible affairs, briefly termed politics, will grasp his indeed only theoretical thought: that politics which genuinely strive to surmount blocs and the division of the world into blocs and to make a further step towards progress of (self-managing) society, that this politics, precisely because they are based on self-management, can be nothing else but socialism, nor should they be anything else but parliamentary socialism. Not private ownership of powerful means of production like in the West, nor political monopoly of a powerful organization like in the East. For no one has in his possession a single and ultimate truth. Every person should state his own truth. Of this the writer is firmly convinced. In the name of that conviction, he is prepared to renounce to many things contained in the lines he has written as delusions, respecting with discipline even the argument of coercion for lack of any better one. There is one thing the writer is not however able to renounce to: that potatoes can be peeled in a different way too, that a wrongly or correctly peeled potato will depend on the circumstances, and that those circumstances are made up of all people. All people, whether they are silent or whether they believe in God or in Matter — or above all in them own selves.


translated by Spomenka Ninθiζ-Šoζ)

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