Mathematical expression and application of Marxism

The word ‘philosophy’ comes from the Greek language and it means ‘the love of wisdom’. There are hundreds of definitions dealing with what philosophy is, but it is generally believed that philosophy is the theoretical and systematic understanding of nature, society and thinking, and it is the ideological system that grasps the entire world as a whole. Chinese philosophy, Western philosophy and Indian philosophy are the three major traditional philosophies in human history; Chinese traditional philosophy includes three factions, Confucian, Taoist and Buddhist. The natural view of ‘the unity of man and nature’, the dialectical view of ‘the unity of yin and yang’, and the practice view of ‘the unity of knowledge and action’ are not only full of the wisdom of ancient Chinese but also establish an internal connection between the ‘unity’ of philosophy and the ‘1’ of mathematics. The Daoist theory posits: ‘The Dao gave birth to one, one gave birth to two, two gave birth to three, three gave birth to all things’ [1], but also agrees with the mathematical expression of the classic philosophy. Western philosophy (including philosophy) originated in Greece in the 6th century BC and mainly discusses the causes behind the phenomena of things. The Marxist philosophy born in the 1840s, namely dialectical materialism and historical materialism, is generally regarded as a revolutionary change marking the development of philosophy. Of course, there are different opinions. For example, the British philosopher Bertrand Russell argued that Karl Marx, as a pure philosopher, ‘had serious shortcomings’ because he was prone to ‘overemphasise reality’ and limited himself to the earth and human beings, and ‘was not qualified to call his philosophy a philosophy of science’ [2]. Practice has proved that this disagreement can only be a misunderstanding, because Marxism is the theory about ‘human liberation’; Marx said of the interrelationship between nature and the world that it is ‘humanisation of nature and the world,’ although what Bertrand Russell referred to as ‘natural in the world’ for Marx, and ‘human liberation’, are largely not related, and also have no meaning; as Marx pointed out, ‘In different ways do philosophers interpret the world. The problem is changing the world.’ This is the fundamental difference between Marx and other philosophers. The fundamental problem of philosophy is the problem of the relationship between matter and consciousness, or existence and thought.

The term ‘Economics’ seems to have been formulated only after the creation of slave society. The publication of ‘Economy’ by Xenophon, an ancient Greek thinker, shows that economics has emerged in the West. Later, another famous thinker in ancient Greece, Aristotle, and Italian scholar Aquinas Thomas etc., put forward some economic theories to defend the slave economy and feudal economy. But economics at that time did not have an independent discipline. With the emergence and development of capitalism, political economy gradually formed an independent social science. Marx said that political economy, as an independent science, came into being in the handicraft industry period. Political economics can be divided into broad sense and narrow sense according to different research objects. The narrow political economy takes the capitalist mode of production as the research object and the broad political economy is a science that studies the conditions and forms of production, distribution, exchange and consumption in various human social forms. Marxist political economy belongs to broad political economy and the goal is to ‘find out the universal laws of production, operation, development, extinction and replacement that can be applied and followed by all social and economic forms, modes of production or production relations.’

Both Cihai and Modern Chinese Dictionary define ‘mathematics’ as ‘the study of spatial forms and quantitative relations in the real world.’ [3] (P. 1212). Academician Li Daqian believes that the ‘real world’ here can also be called ‘the real universe, the material universe, the material world and the objective world’. So, mathematics is the science of the shapes and numbers of the real universe (the real world) [4]. Specifically, the main research space, structure, quantity, change and information such as concept, are divided into higher mathematics and elementary mathematics. ‘What is Mathematics’ co-written by the world's outstanding mathematician R·Courant (Richard Courant) and the famous statistician H·Robin Herbert Robbins in the 20th century (later, a chapter was added by Ian Stewart, a professor of mathematics at the University of Warwick) summarises the content of mathematics into arithmetic and algebra, analytic geometry, geometric drawing, projective geometry and non-Euclidean geometry, topology, function, limit and continuity, maximum and minimum, calculus, integral method etc. Mathematics is widely used not only in people's daily work and life, but also is an extremely important and irreplaceable tool for scientific research. As Engels first pointed out when enumerating various sciences and their dialectic content in the ‘Dialectics of Nature’, ‘Mathematics is an auxiliary tool and form of expression for dialectics’ [5]. Karl Marx pointed out more profoundly, ‘A science can only be regarded as a true science when it can successfully use mathematics’ [6].

There is a great relationship between political economy and mathematics. Whether it is general political economy or political economics, it can be regarded as a quantitative relationship based on the four basic links it contains.

In terms of methodology, Mao Yiran gave a systematic elaboration in ‘On the Relationship between the Methodology of Political Economy and Mathematics’, pointing out that ‘in terms of general methodology, mathematics is general to the methodology of political economy. The reason why the study of political economy should apply mathematical methods lies in the quantitative relations of the research objects themselves.’ [11] In fact, the development of modern science and one of the important characteristics is the social science and natural science, no matter in theory or on the way of mutual fusion and penetration. When political economics is studying social phenomena, it is necessary to study qualitative regulations and changes, as well as quantitative regulations and changes, which must rely on mathematical methods. For example, the input–output method in economic management must use mathematical methods.

Karl Marx loved mathematics all his life and made outstanding achievements. Karl Marx regarded the study of mathematics as an important source of rich materialist dialectics. After deep study, he believed that in higher mathematics, he had found the most logical and the simplest form of dialectical movement. This is especially true in political economy, where his economics notebooks are filled with algebra, mathematical formulas and graphics. The most direct example of Karl Marx's talent and diligence in mathematics is his Mathematical Manuscript [13]. Karl Marx in such aspects as differential has obtained a high achievement; not only did he study it himself but he also wrote to persuade Engels to study calculus; later, Engels said in his reply to Marx on August 18, 1881 about the discussion of derivative functions, ‘this matter caused me such great interest that I not only consider the whole day, but also dream, looking at it’ [14]. We know that the great philosopher Hegel's mathematics knowledge is extremely abundant, so that none of his students can sort out and publish the large number of mathematics manuscripts he left behind. But Engels wrote about the matter, ‘As far as I know, the only person who knows enough about mathematics and philosophy to do this job is Marx’ [15]. Once Karl Marx's paper on differential calculus was published, it caused a great sensation in the international academic circles. In 1977, Kennedy, a famous American scholar, gave an academic report entitled ‘Karl Marx and the Foundations of Calculus’ at the International Conference on the History of Mathematics held in Germany (then West Germany). It can be seen that Karl Marx's achievements and accomplishment in mathematics were substantial.

According to the above analysis, Marxism mainly includes philosophy, political economy and scientific socialism. As far as its relation with mathematics is concerned, it is prominently manifested in philosophy and political economy. In terms of Marxist philosophy, in addition to the relationship between philosophy and mathematics applicable to the previous analysis, it is mainly manifested in Marxist philosophy, that is, the scientific world view of dialectical materialism and historical materialism, which plays an appropriate guiding role in mathematical research. As Engels pointed out, ‘the concept of number and form is not from anywhere else, but from the real world’ [16]. In ‘On the Relationship between Mathematical Education and Karl Marxist Philosophy’, Jiang Jianlin comprehensively discussed the significance and guiding role of Karl Marxist philosophy in mathematics teaching and research from the perspective of mathematics education.

The relationship between Marxist political economy and mathematics is even more obvious. ‘In formulating the principles of political economy,’ Marx wrote in a letter to Engels on January 11, 1858, ‘I have been greatly hindered by errors in mathematical calculations. I have been disappointed and have had to sit down again and go over algebra quickly’ [17]. Marxist political economics focus on material production, start with the study of goods, through goods production, distribution, exchange and consumption, and also reveals the source of surplus value produced with the secret. Reading ‘Das Kapital’, we will find that Marx uses a lot of mathematical knowledge and mathematical formulas for calculations. For example, when studying the form of value, he starts with the simplest mathematical equation like ‘1 sheep = 2 axes’, and then discusses the political and economic essential meaning behind the equation. When studying the law of currency circulation, many mathematical formulas are also used, such as ‘the amount of money needed to perform the functions of the means of circulation = the total price of commodities/the circulation speed of the same unit of currency’, and so on. This kind of mathematical expression is even more numerous when it comes to more complex political economic concepts or principles such as the organic composition of capital, the rate of profit and the rate of surplus value. Later, scholars also made many achievements in the research of the mathematical model of Marxist political economy and examples of these are Wu Yifeng's ‘Research on Mathematical Models of Marxist Economics’, Feng Jinhua's ‘Mathematical Principles of Marxist Economy’, Ma Yan's ‘Mathematical Analysis of Modern Political Economics’ etc. Scholars such as Ding Baojun and Bai Baoli have achieved a number of research results in the specific field of Marxist political economy [18].

Both Marxism and mathematics arise from human production and labour and social practice. The two have an inseparable internal connection. Marxism provides a scientific world view for mathematics research, and mathematics provides a scientific methodology for Marxist research. ‘Marxism accommodates the theory of mathematics with a broad mind, and mathematics has enriched Marxism with extensive and profound knowledge. Marxism has played a huge role in the development of mathematics. At the same time, the development of mathematics has also greatly promoted Marxism, especially the development of Marxist economics; mathematics and Marxist philosophy promote each other and develop together’ [19].

Karl Marxist philosophy insists that matter determines consciousness and consciousness has active effect on matter. The former belongs to ontology, while the latter belongs to epistemology. These philosophical ideas are very important, but difficult to understand, but they can be greatly assisted by mathematical expression. As Engels once pointed out, ‘The fact that phenomena of different natures have the same mathematical expression form shows that phenomena of different natures can have the same quantitative relationship, reflecting the unity of the material world’. This relation can be expressed as follows:

‘Material’ is equivalent to the independent variable of the function, denoted by M; ‘Consciousness’ is equal to the function of the dependent variable, C; according to the relationship between the expression of C = f (M), it indicates that the on the ontology, consciousness is determined by the material, what kind of material is objective existence, will have what kind of consciousness, in epistemology, that is for material that is objective existence of things, And at the same time it shows that consciousness can know the objective world.

The above mathematical expressions on the ontology and epistemology of Marxist philosophy clearly show that without matter M, there is no consciousness C. C is the processing of M by the human brain. Therefore, in the ideological line, to adhere to Marxism is to insist on proceeding from reality in everything and seeking truth from facts. M is ‘facts’, ‘seeking’ is f(M) and ‘truth’ is C, whereas the correct f(M) gives rise to the process and results.

Another important category of Marxist philosophy is the mathematical expression and application of the principles of universality and particularity. The principle of Marxist universality and particularity reflects the relationship between commonality and individuality of things. Universality refers to the commonality of similar things while particularity refers to the characteristics of similar things that are different from other things. Commonality inevitably includes individuality, and individuality is inevitable. On reflecting the commonality, one must ascertain for himself that understanding the commonality must start from the individuality of the object. Engels pointed out that ‘all true and detailed knowledge is only in: we raise individual things from individuality to particularity in our thoughts, and then from particularity to universality’ [3]. The constructive function relationship in mathematics is the process of rising from personality to commonality. Once the relationship expression is established, it means that the commonality and personality relationship in similar things has been found, and it also means that a certain aspect of the law has been found. The ‘special value solution method’ is often used in mathematics to solve the problem through the special ‘personality’ that must meet the ‘common’ of the same kind.

Known, for example, that a, b, c are real numbers, and for any real constant has |x+a|+|2x+b|=|3x+c|, so a: b: c = _____. To solve this problem by identity, the key is to find a, b, c and the relationship between the three real numbers; if there is no special method to solve, this would be more troublesome, but according to the question, we can meet the requirements of using special values, ‘for any real number x’; |x+a|+|2x+b|=|3x+c| equation was established, given the commonness, so as long as the real numbers were found to be in line with the conditions; and to facilitate problem solving with the similar personality, x equations must be set up, so that it is easy to find a, b, c and the relationship between the three real numbers. The details are as follows: Solution: Because c for real, So can make x=−c/3, Then |x+a|+|2x+b|=0 ∴ x=−a, x=−b/2 ∴ −c/3=−a=−b/2 ∴ c=3a, b=2a ∴ a:b:c=a:(2a):(3a)=1:2:3. \matrix{{{\rm{Solution}}:\;{\rm{Because}}\;{\rm{c}}\;{\rm{for}}\;{\rm{real}},} \hfill\cr{\quad \quad \quad \quad \,\,{\rm{So}}\;{\rm{can}}\;{\rm{make}}\;{\rm{x}} =- {\rm{c}}/3,} \hfill\cr{\quad \quad \quad \quad \,\,{\rm{Then}}\;|{\rm{x}} + {\rm{a}}| + |2{\rm{x}} + {\rm{b}}| = 0} \hfill\cr{\quad \quad \quad \quad \,\,\quad \quad \therefore \;{\rm{x}} =- {\rm{a}},\;{\rm{x}} =- {\rm{b}}/2} \hfill\cr{\quad \quad \quad \quad \,\,\quad \quad \therefore \; - {\rm{c}}/3 =- {\rm{a}} =- {\rm{b}}/2} \hfill\cr{\quad \quad \quad \quad \,\,\quad \quad \therefore \;{\rm{c}} = 3{\rm{a}},\;{\rm{b}} = 2{\rm{a}}} \hfill\cr{\quad \quad \quad \quad \,\,\quad \quad \therefore \;{\rm{a}}:{\rm{b}}:{\rm{c}} = {\rm{a}}:(2{\rm{a}}):(3{\rm{a}}) = 1:2:3.} \hfill}

It can be seen that philosophical thinking is quite important to solve mathematical problems. China's annual college entrance examination math questions often set such questions and the purpose of the examiner is mainly to investigate the candidates’ philosophical thinking.

The law of surplus value, known as one of the two great discoveries of Marx, is the cornerstone theory that reveals the nature and secrets of capitalist exploitation. It is more difficult to understand if it is only expressed in words. If Marx's profit rate formula is slightly deformed, it will be much easier to understand, that is to say, it will be easier to understand and see the exploitative nature and secrets of capitalism. According to the definition of profit rate in Marxist Political Economy, P′ represents profit rate, M represents surplus value, C represents constant capital and V represents variable capital; then, profit rate can be expressed by the following formula: P′ = M/C + V.

In The Application of Mathematical Expression in the Teaching of Karl Marxist Political Economy, Xia Shuilong made the following transformation and analysed the formula: divide the numerator and denominator of the right end of P′= M/C + V equation by V at the same time, and we can get: (1) P'=m/Vc/V+V/V=m1c/V+1 {P^{'}} = {{m/V} \over {c/V + V/V}} = {{m1} \over {c/V + 1}}

It can be seen from the deformed Eq. (1) that if C/V is certain, then (C/V) + 1 is also certain, then P′ is proportional to M′, that is to say, the higher the rate of surplus value, the higher the profit rate will be, and vice versa. This is the effect of the rate of surplus value on the rate of profit.

If M′ is constant, then P′ changes in the opposite direction to C/V; and C/V represents the size of the organic composition of capital. Obviously, the higher the organic composition of capital, the lower will be the profit rate, while the lower the organic composition of capital, the higher the profit rate will be. This is the effect of the organic composition of capital on the profit rate. It can also be seen from Eq. (1) that, when other conditions remain unchanged, i.e. M′ and V remain unchanged; C is the denominator of the equation, so C decreases, the value of the denominator decreases and P′ increases, that is, constant capital saving can increase profit rate. This is the effect of constant capital savings on profitability [20].

From the above analysis, it is not difficult to see that the mathematical expression of some principles of Marxism, or the appropriate transformation of the original formula, will be of great help to our analysis and understanding of Karl Marxist principles.