MATHEMATICS AS AN INDISPENSABLE TOOL.
Mathematics is an efficient and necessary ‘tool’ for the study of other subjects. It is literally indispensible for the study of subjects such as physics, chemistry and astronomy, and no informed person would question its instrumental value in this connection. This can be illustrated by showing the dependence of various subjects upon mathematics.
Mathematics and the Physical Sciences
Many people believe that modern civilization is only a product of the sciences and technology. They regard mathematics as the domain of absolute truths; and hence contend that it has nothing to contribute to civilization. In fact, mathematics is the ‘tool’ of the sciences. “quantitative measurements and their mathematical manipulations are….. woven inextricably into the structure of sciences. Within large area of physics, chemistry and biology one can no more exercise mathematics than logic without destroying the essential structure”. “General Education in a Free Society” p.152.
Water when heated produces steam, and steam exerts pressure on the surface of the vessel in which it is confined. This is only a qualitative property. This alone does not constitute a complete knowledge of steam. We must also know a good deal of the quantitative side of the properties of steam, – that is the amount of pressure that a particular quantity of steam under definite circumstances will exert on the surface of the vessel. From advancing from qualitative properties to quantitative relations, mathematics is an essential tool. Similarly, the knowledge of the deflection of ray of light entering from one medium to another of different density is incomplete, and less useful for purposes of making eye glasses and optical instruments, unless the quantitative law of refraction is also known. In short, to understand the secrets of nature, and to bring her to the service of man, the quantitative aspects of physical phenomena must be known, for which mathematics is an indispensable tool.
In the experimental sciences, in order to derive fundamental truths, one has to collect the empirical data, organize and analyze them systematically, interpret the conclusions and extract fundamental truths. For all these purposes, knowledge of statistical methods, calculus, differential equations and vector analysis as well as the ability to determine an empirical function, which will suitably represent the given data, is essential. Indeed, mathematics is “a symbolic language, whose practical function is to express quantitative and special relationship”
Interpretation of mathematical symbolism applied in the physical sciences often leads to new results in scientific fields. Many scientific theories have come into being only because the very ideas which give meaning to these theories dawned in the minds of mathematicians years before anyone predicted their possible applications to the sciences. The field of science is replete with instances to illustrate this fact. For instance, many of the properties of electromagnetic waves were deduced from Maxwell’s equations, even two decades before they were realized in the laboratory by Heinrich Hertz, a German scientist. Maxwell found out, by mathematical reasoning, that waves take some time to travel. He calculated the rate of travel and the length of the waves. He also found that waves would go slower in glass and water than in air; they would curve round sharp edges; they would be partly absorbed and partly reflected by conductors and so on.
The geometry of Reiman, which in fact, he developed half a century before the time of Einstien, has led to Einstein’s theory of relativity. The general theory of relativity according to which Einstein predicted that light from the stars passing through the gravitational field of th sun would bend inwards towards the sun on its way towards the earth, and Einstien’s calculation of the degree of deflection were proved to be true by physicists only years after because only when (i.e. in 1919) they were able to make direct observations of the sun during a solar eclipse. The discovery of various ‘elements’ whose existence were predicted by the ‘periodic law’ and of the planet Neptune whose position was predicted by as-tronomical calculations, are other examples of he fact that the major scientific theories have come into being only because the very ideas which gave these theories meaning were created by the mathematicians years, decades, or even centuries before anyone foresaw possible applications to science. Thus we see that mathematics is a means not only for giving precise and economical statements for truths previously expressed less accurately and gracefully but also for affording new methods of discovery, in some instances excelling the resources of the laboratory, just because it can bring one to grips with basic considerations.
Thus, in the foregoing paragraphs we have seen that mathematics is responsible for many discoveries.
Mathematics and Biological Sciences.
The dependence of biology upon mathematics can be understood from the use of apparatus like monometer for measuring root pressure, potometer for measuring the rate of transpiration, and auxanometer for measuring plant growth as well s from the graphical representation of the chemical composition of a plant body of the growth of a plant during a specific period of assimilation and absorption and the like that are found in any textbook dealing with the subject. Only in recent years mathematical principles and methods have been applied in biological experiments. No wonder such special phases of biological and physiological study as genetics, heredity, nutrition, metabolism (anabolism and catabolism) and fatigue have advanced remarkably in the twentieth century.
The relationship between biology and mathematics is two-fold: direct and indirect.
The direct relationship is seen from the wide use of mathematical symbolism and mathematical methods. As in every other branch of science, in the field of biology also the work of the biologists consists in the direct observation of natural phenomena, classification and comparison of observed facts, and derivation of a general law. At every stage of these processes, mathematics is of immense help.
Biophysics and Bio chemistry (these branches are becoming increasingly important in understanding the nature and effects of radio activity, X-Rays, Ultra violet light and the Electronic microscope) explain the physical and chemical changes in the living cells are the two branches of biology that have recently developed as independent sciences. Mathematics is partly responsible for the progress of these studies. Therefore, biologists have to master biomathematics.
Life process is complex and complicated. It implies that biological truths cannot be always derived through elaborate experimental methods. Statistical methods are often found to be essential. In the study of the measurement of stature, Calton applied statistical methods. He selected a number of tall fathers and calculated their mean height and that of their sons. He found that the mean height of the sons is not equal to that of the selected fathers, but is rather nearer to the mean height of the population as a whole. This he expressed by saying that the mean height of offspring tended to ‘regress back towards the mean of the race’.
As in the case of physics, in biology also mathematics is an indispensable tool not only to describe experimental results in concise and clear language but also to interpret, discover and to formulate general laws. Perhaps in the study of biological phenomena one cannot go very far beyond the descriptive stages without the aid of mathematical analysis and mathematical treatment.
In order to show the indirect relationship, we may consider the following problems:
In problems that entail indirect measurement, or quantitative phenomena, we come across such mathematical terms like transpiration co-efficient, which have to be calculated through mathematical manipulations and not through direct measurement. Mathematics contributes more or less to the same extent to the other fields of science also. There is much truth in this: “mathematics is the gate and key of the sciences.”
Mathematics is also a tool subject for the learning of the social sciences, such as economics, sociology, psychology, philosophy and logic. These social sciences also draw heavily upon mathematics, particularly statistical methods, graphical representation and the methods of calculus for investigations as well as for the interpretation of social phenomena. This can be illustrated in the case of Economics.
The work of an economist, like that of a physicist or of a biologist, consists in the collection of data describing a social phenomenon and in the interpretation of the data to discover general truths. In these processes statistical methods are widely used. The use of graphical representation for the presentation of data in economics, of the theory of probability for the calculation of the expectation of life in connection with life insurance, and of the formulae for things such as capitalization and fixing annuities, premiums, and sinking funds, etc., is well known. As mathematics is essential for the study of economics, it forms an integral part of degree courses in economics in many universities in India and abroad.
Obviously, mathematics is mainly concerned with establishing relationships such as A=P (1+r/100) n. (It gives the amount A of the principal P invested for n years r compound interest) between variables and between functions. Economics too deals with capital, labour and taxation etc., under certain conditions. Demotingthese ‘attributes’ by letters their relationships can be found in the form of functions. For instance, if D represents demand (quantity sold) and P the price of an article, then D =f(P), represents the demand for the article, which is a function of the price, provided other things are equal. Thus, we see that mathematical language and methods are important in describing and expressing general laws in mathematics.
Another important use of mathematics is, from the function relating to attributes, many other relationships can be derived. Application of mathematical principles to economics has enabled economists to find out new relationships, and hence to discover new hypotheses. In the above illustration, we can get the value of P which will give the greatest profit to the producer; that is when P.D. is the maximum d/dp (P.D)+0 and d2/dp2 (P.D) is negative. Similar illustrations are profusely found in any look on mathematical economics.
Irving Fisher of the Yale University has tried to show that” it is possible to estimate numerically, through suitable mathematical equations, the velocity of circulation of money as a function of the variables such as the volume of money in circulation, the annual flow of money into and out of banks, and the annual cash payments to labour”. (The Sixth Year Book, The National Council of Teachers of mathematics.1907, p.12f ). Infarct, he has shown that “money in the United States circulates about twenty five times a year. In other words, the average dollar stays in the same pocket about two weeks”.
Mathematics is useful to express laws in economics tersely, clearly and concisely. The law of Diminution of Marginal Utility is an illustration to the point. Dr. Marshall defines it as the ‘additional benefits which a person derives from a given increase of his stock of anything and which diminishes with the growth of the stock that he already has’. This can be empirically expressed by the facts, that Marginal Utility is measured by du/dx andd2u/ dx2 is negative, where ‘u’ is the total utility of an amount ‘x’ of a commodity to a person, at a given time. So much so, Marshall compliments mathematicians and says that the direct application of mathematical reasoning to the discovery of economic truths has of late ‘rendered great service in the hands of master mathematicians’.
Thus, mathematics plays an important role in studying other subjects.
References
1. General Education in a Free Society , Report of the Harward Commission, Cambridge(Mass):
Harward University Press, 1945. P.152 f.
2. Roger Bacon, As quoted in ; N.Kuppuswami Iyengar, op.cit., p,8, pp.25.
3. The Sixth Year Book, The National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University,1907, p.1 f.