### Summary

The current paper is devoted to the modern trend of the mathematization of economics and the role of Willian Stanley Jevons in it. The relevance of mathematization in economics is defined, and the place of mathematics in economics is determined. The paper provides positive and negative aspects of the mathematization of economics and the reasons for its use. Moreover, the importance of mathematical economics and existing concerns regarding its excessive use are outlined within the given research.

The paper gives the analysis of arguments concerning the mathematization of economics that are presented by the representatives of different schools of economic thought and specifies Jevons’ views regarding this notion. Furthermore, there is an outline of the historical development and applicability of mathematical methods. The impact of the mathematization of economics and the role of William Stanley Jevons in this current trend are determined. In conclusion, the paper provides the idea that mathematics, logic, and descriptive techniques may be used as complementary tools.

## The Relevance of Mathematization in Economics

Mathematization is currently present in almost all spheres of economics. This trend may be observed from the resolution of World War II, and it has become even more intense at the end of the 20th and the beginning of the 21st century (Debreu 1). The mathematization is relevant to economics in a number of ways. First of all, it increases the applicability of economics. It is possible to use economic concepts in relation to a large variety of social and statistical issues. Moreover, mathematization contributes to higher accuracy of actual results and policy recommendations. It is possible to quantitatively describe the dynamics of main economic parameters and processes including supply, demand, savings, investments, consumption, GDP, etc.

Jevons also proposed to use mathematical methods for explaining the relationships between consumption and labor decisions (Jevons 310). In fact, the concept of marginal utility in the tradition of Walras and Jevons is based on the maximum possible use of mathematics in economics. It may be attributed to other spheres of interest of these scientists, as Jevons was interested in chemistry at the beginning of his career. The approach of Carl Menger was different. He was in favor of economics as a more descriptive and logical science. However, the position of Jevons has appeared to be more influential on mainstream economics. Mathematical methods are widely applied nowadays, and new economic disciplines that use mathematics are constantly emerging. In particular, econometrics is extremely popular all over the world. It may serve as an example of the mathematical influence on economics. The mathematization of economics is one of the most characteristic features of the present period of economic science development. Although it has some negative aspects, this tendency seems to be objective.

### The Place of Mathematics in Economics

Different scientists express different views regarding the role and place of mathematics in economics. The majority of them recognize that modern social science cannot demonstrate significant growth without the implementation of at least some achievements of mathematics and statistics. However, different schools of economic thought have different positions in relation to those fields that should be analyzed with the help of mathematics. Neoclassical economists agree with Jevons that mathematics may be applied in almost all spheres of economics. Jevons proposed to use mathematical methods in order to determine the comparative resources of different countries; he built his analysis using the coal market as an example (Jevons 158). Thus, Jevons considered that mathematics might be applied to both micro- and macroeconomic issues. The same ideas are expressed by neoclassical economists and Keynesians. The only difference is that neoclassical economists are mainly interested in microeconomics, while Keynesians are primarily concentrated on macroeconomics.

The representatives of the Chicago School of economics express other views. They use the partial equilibrium model rather than the general equilibrium one. However, they also widely apply different mathematical methods including the regression analysis. Although they suggest that all markets should be analyzed separately and only then integrated into the whole picture, these scientists agree that mathematics may be highly useful for deep economic research. Thus, all these schools of economic science suggest that mathematics should be widely used in almost all spheres of economic research. However, the representatives of the Austrian School of economics believe that mathematics should not be applied in any social science including economics. They suggest that economics should be more realistic and logical while the majority of econometric models cannot objectively represent the existing state of the economy. According to this view, the place of mathematics in economics is moderate.

### Positive and Negative Aspects of the Mathematization of Economics

Although the mathematization of economics is an objective tendency, it has both positive and negative aspects. All major positive aspects were correctly predicted by Jevons. The economic science has greatly increased its sphere of applicability. Moreover, economic experts are able to cooperate with scientists from other disciplines as they have a common method of studies. With the help of mathematization, it is possible to use economics as a science for making social predictions. Previously, economists were able to provide only qualitative analysis. For example, they could say that if the market demand increased, the price would rise as well. However, it was impossible for them to specify the exact percentage of the increase in the market price. Nowadays, the use of mathematical and statistical methods allows making reliable predictions. Another positive aspect refers to the fact that regression analysis enables understanding the relationships among those factors that are not directly logically related. Their significance may be determined, and the corresponding strategy may be implemented.

However, a number of negative aspects are present as well. First of all, mathematical methods cannot lead to the same effective results that may be observed in the sphere of natural sciences because there are no constant or fixed parameters in any of the social areas. Technologies and consumers’ preferences may constantly change, and the regression analysis cannot adequately represent the human ability to learn and adapt one’s behavior to the external conditions. As individuals do not react mechanically to external impulses, the essence of human reason and consciousness should be taken into account. Therefore, economics should not be treated as a purely mathematical science; it should be based on logical conclusions as well. The classical logical analysis may be useful in a number of cases where there is a shortage of quantitative information. Moreover, mathematical calculations may be verified with the help of logic.

### The Reasons for the Use of Mathematics in Economics

It is reasonable to outline specific reasons that demonstrate the usefulness of mathematical methods in economics. As the same events and processes may be presented in different ways, the mathematical approach allows receiving one more dimension of investigation. It is possible to receive additional information regarding a large number of problems under investigation if quantitative aspects are taken into account. Moreover, the essence of the majority of economic aspects is such that it should be quantitatively evaluated. For example, the issues of inflation and unemployment cannot be fully analyzed exclusively from the qualitative perspective. It is necessary to understand the dynamics of economic parameters in order to determine the corresponding policy for solving specific problems. Without the developed system of statistical measures, it is impossible to examine the situation in any industry of the economy.

In fact, the whole system of government interventions and regulations is heavily based on the use of mathematics and statistics. The government may develop its policies and try to forecast its impact on the selected dependent variables. For example, the government may try to decrease unemployment in the country. For this reason, it may decide to increase its spending. However, it may lead to additional inflation as well. In order to assess all positive and negative aspects of its activities, modern econometric models may be helpful. They may also be used as an analytical tool for reliable prediction in relation to all main macroeconomic characteristics. However, as Jevons pointed out in various works, the analysis should be based on marginal units. In other words, only if the expected marginal revenues are higher than the expected marginal costs, the program under consideration should be introduced. Otherwise, it should be rejected, as there is another (more profitable) alternative option for the use of available resources. At the same time, the marginal approach should not intervene with the analysis of structural problems (Hudson 292).

### The Importance of Mathematical Economics

The importance of mathematical economics may be observed in different fields. First of all, it is important in relation to the complex examination of situations and existing economic phenomena. Moreover, it is significant because it helps to determine quantitative relationships among different economic parameters. For instance, it is possible to predict what additional percentage of inflation may be expected in the economy if the level of unemployment decreases by 1%. The interpretation of all modern econometric models is impossible without adequate knowledge of mathematics and statistics.

However, it should be stressed that the number of testable empirical facts is small in comparison with a large variety of mathematical hypotheses (Beed and Kane 583). It means that it is not always possible to verify the applicability of a given model. If the amount of statistical information is inadequate, it may be problematic to construct a reliable model. Therefore, mathematical economics cannot solve all problems, and other methods should be used as well. In order to increase the significance of mathematical economics, additional attention should be paid to the correct interpretation of the results and calculations. Mere facts and calculations per se do not mean any particular policy. They should be adequately interpreted with the use of the most appropriate theories. Therefore, mathematical economics cannot be used in isolation and should be supported by other analytical tools.

### Concerns Regarding the Excessive Use of Mathematics in Economics

Although the use of mathematics and statistics in economics has a number of significant advantages, many experts suggest that some extensive use of mathematical methods in economics may be observed. It leads to the situation when new techniques cannot be effectively used, and the results cannot be correctly interpreted. Moreover, economics is not a purely objective science (like physics, chemistry, or mathematics) as subjective preferences and choices of individuals play an important role. Mathematical techniques beyond simple geometry and algebra may contribute to additional inconsistencies and theoretical misunderstandings (Beed and Kane 581). Although Jevons was in favor of mathematical methods, he used them to a reasonable degree; the majority of his methods were similar to those he used in chemistry (Black 215).

### The Impact of the Mathematization of Economics

The mathematization of economics has revolutionized the whole system of social sciences. For a long time, it was suggested that mathematical and statistical methods could not be used in social sciences because their subject is different. However, it was discovered that mathematical methods might be of great use in economics as well. It is possible to describe economic phenomena with the help of mathematical models and equations. Although these representations of reality are not always perfectly realistic, they may be useful. The ideas of Jevons have significantly influenced the development of mainstream economics including Keynesians (Harrod 15).

As mathematics is a broad science, not all of its branches may be equally helpful in the field of economics. Therefore, only the most suitable spheres should be selected and correctly applied. In particular, the game theory seems to be popular among mainstream economists nowadays. Jevons suggested that the whole production process could be accurately presented in a mathematical form (Jevons 283).

### Arguments Concerning Mathematical Economics

As different schools of economic adopt different methods of study, their evaluation of the role of mathematical economics is different. The majority of mainstream economists suggest that mathematization of economics should be expanded and additional mathematical tools should be introduced in the field of economics. They believe that the trend towards higher mathematization should continue. As mathematics provides an objective analysis, it may serve as a helpful analytical tool for receiving information for subsequent policy recommendations. The economists from the Chicago School of economics also try to use mathematical methods. They are primarily focused on the analysis of specific markets as they have adopted the framework of partial equilibrium in their analysis. They understand the potential threats that may be associated with over-generalization and abstraction of their theories. The representatives of the Austrian School suggest that the use of mathematics in economics leads to negative rather than positive outcomes. They believe that the sphere of human action cannot be adequately described with the help of any mathematical tools or models. Therefore, they propose to use the traditional descriptive approach.

Some scientists suggest that economics should be viewed as a pure theoretical science (Reed 3). Moreover, this position was adopted by Adam Smith. According to this view, mathematics may occupy a very moderate place in economics. In particular, only the basics of algebra and geometry may be useful, while the rest of mathematical studies have no practical relevance to economics. The representatives of this perspective believe that economists should not adopt the new methods from such areas as mathematics or statistics but they should rather be focused on the development of pure economic methods of analysis.

Thus, there are different positions regarding this issue. Those schools and scientists who advocate extensive use of mathematics in economics are generally influenced by the ideas of Jevons and Walras. It seems that mathematical methods should not be considered as the opposition to all other theoretical approaches. It is possible to use both perspectives in the process of examination of particular issues. It seems that different approaches may be optimal in analyzing different economic problems.

### Jevons’ Views Regarding the Mathematization of Economics

William Stanley Jevons is one of the most influential economists of the 19th century. His contribution to the current trend of the mathematization of economics is evident. Jevons understood that the marginal approach might be correctly presented in a mathematical form (Zabell 172). Therefore, he proposed to measure the value of goods with the help of marginal units. His initial analysis was primarily related to the labor market, but after some time, it was further extended to other areas. The marginal approach created the foundation for a reliable economic investigation of different issues. It may also be used as a basis for rational decision making. In fact, almost all current optimization problems in economics are solved with the help of mathematical methods.

The analysis of utility (even on the national and international levels) implements the approach developed by Jevons. He has proved that the value of goods does not depend on the costs of labor (Schabas 65). On the contrary, the value of final goods influences the value of factors of production. Therefore, the whole system of Marxism was refuted after the discovery of Jevons, Menger, and Walras. In general, Jevons was sure that mathematical methods might be very helpful in the study of economics. Even in the 19th century, it was evident that a large number of economic problems should be analyzed with the help of quantitative methods. However, it was difficult to determine the exact algorithm of such methods. The marginal approach has demonstrated that if the value of marginal units is taken into account, then the rational decisions may be made. Jevons understood the importance of this fact and believed that mathematical methods could be very effective in this context.

Jevons also suggested that economic experts should be able to make quantitative predictions. Previously, they were able only to determine the general direction of changes. For instance, if the market supply decreases, the prices will tend to increase. However, they could not forecast the exact percentage of change. However, this information could be of main interest to business people for rational decision making. Therefore, Jevons created a system that was able to address efficiently the real needs of entrepreneurs.

### History and Applicability of Mathematics in Economics

Economic issues were of some interest even for people from Ancient Greece. However, the science formally emerged only in the second half of the 18th century after the systematic treatise “Wealth of Nations” by Adam Smith. Smith used the descriptive approach that was widespread at that period. It seemed that it was possible to describe the whole set of economic principles without the use of mathematics and statistics. Some statistical information could be used only as an illustration of some theoretical conclusions. No special statistical or mathematical methods were adopted.

The situation changed when Jevons, Menger, and Walras developed the concept of marginal utility. Jevons and Walras concluded that economics should be based on the use of mathematical techniques while Menger believed that marginal utility could be analyzed with the help of the traditional descriptive approach. Thus, the development of mathematics and statistics in relation to economics was primarily influenced by Jevons and Walras. However, there is a strong debate among historians and economists regarding the process of mathematization (Lee 73).

Some mathematical methods may be applied in the sphere of economics because both experts and business people need the foundation for the objective analysis of empirical data. With the help of mathematics, it is possible to systematize available information and use it for forecasting. The concept of marginal utility helped to resolve the classical paradox of value (Mirowski 369). It became possible to explain why the value of gold was high while the value of water was low. Jevons has demonstrated that the answer refers to the fact that the value of all goods is dependent on the marginal rather than total utility. It helps to compare the value of different goods and use this information for subsequent calculations.

Jevons has demonstrated that mathematical methods may provide the most reliable information in relation to all main spheres of economics. Economists understood that they could not only describe the facts of economic reality but also efficiently intervene with their development. Modern econometric models are typically constructed for the reason of effective intervention regarding economic aspects under consideration.

### Conclusion

The current development of economics is impossible without the excessive use of mathematics. It has created a number of opportunities for experts. First of all, it is possible to receive objective and reliable information regarding the facts of economic reality. Moreover, it is possible to construct models with different strategies of intervention. Forecasting is also based on the use of mathematical methods. This trend towards mathematization originated after the Marginalist Revolution in the 1870s. One of the main contributions was made by William Stanley Jevons.

However, excessive use of mathematics may lead to some negative consequences as well. Not all econometric and statistical models are realistic and able to make reliable predictions. Therefore, it seems reasonable to use logic and traditional approaches in order to correctly interpret information that may be received after the realization of mathematical methods. Mathematics, logic, and descriptive analysis should not be viewed as mutually exclusive methods.