F W Levi: Why Mathematics?

F W Levi delivered the Presidential address Why Mathematics? at the Thirteenth Conference of The Indian Mathematical Society held at Annamalainagar in 1943. F W Levi was at that time the Hardinge Professor of Higher Mathematics, Calcutta University. The Address was published in The Mathematics Student 11 (1-2) (1943), i-ix. It was also reproduced in C S Aravinda and V Remmert, Friedrich Eilhelm Levi in India, Bhavana 8 (2) (2024). We give a version below.

Why Mathematics? by F W Levi.

Even in this age of restless work and incessant labour, thoughtful human beings have the privilege - and even the duty - to lay down their tools from time to time and to think over the true meaning of their task and of their work. Far from the focus of public interest, neither much encouraged by the applause of the masses nor disturbed by their criticisms, mathematicians enjoy an unusual intellectual liberty which implies the full responsibility for their attempts and for their achievements. We are enthusiasts of our science; we enjoy its harmonious beauty and the triumphant moments of successful thinking. Otherwise we could not be mathematicians. But are we, as responsible persons, justified in devoting our whole life and our full energy to these activities? Is mathematics done only for the enjoyment of its disciples, or is it a necessary part of human culture? Why Mathematics?

I could not raise this question at today's festive occasion without believing in the importance and necessity of mathematics, but I ask the audience not to take the question mark as a mere rhetorical figure, some doubts and scruples uttered at the beginning of the speech to let the subject appear even more radiant in an apotheosis at the end of the address. The question is asked in full earnest; the answer, far from being complete, is only an attempt to give some clues to what seems to me the foremost arguments justifying mathematics as a science and the principal reasons for the importance of its further advancement. For this investigation we have to try a descent to the depth where the roots of science are to be sought, but let us start from the surface.

In general a mathematician is as profoundly convinced of the importance of his subject as are e.g., painters, sculptors, musicians and actors. A painter, asked why he paints, may answer that he feels compelled to do so and he may refer the questioner to the books on the philosophy of art stored in the libraries to gratify his theoretical curiosity. This attitude is sincere and respectable though it does not involve a satisfactory reply to the question. A mathematician has an easier way to face - or at least to escape - such a challenge. "Mathematics" he may say "is a complex subject, but its usefulness is evident by its applications. Without mathematics there would be no railways, steamers, motorcars nor aeroplanes. The production of our industrial age is based on mathematical calculations and the administration needs mathematical applications of statistical data. Thus mathematics is an essential prerequisite of modern economic life" - There is nothing untrue in a reply of this kind, but nevertheless it is insincere because it avoids the main issue: the value of mathematics in itself. The boasting reference to the successes of applications of mathematics is a convenient expedient to avoid a superficial conversation with frivolous people about a serious topic, but even from a purely practical point of view there is some danger in it. It is not good to burden mathematics with the responsibility for modern life and all its consequences which show many embarrassing features. People may argue that the destruction of mathematics may bring about the fall of the whole structure of modern life which is based upon it. It is not so long ago that the progress of science and engineering made the public expect the approach of a golden age in the future, perhaps even in the near future. Philosophers and other observers who were horrified by certain cultural developments of that period, were disregarded, but in later years events and changes in political, social and economic life reduced and eventually shattered the hope of a golden age due to scientific and industrial uplift; again people looked back to a golden age of simplicity and safety at a time unknown to the historians, since history records only blood, sweat and toil - and the longing for a golden age. "Beatus ille qui procul negotiis ut prisca gens mortalium" (Blessed who is far from business as were people at some remote time); this ode of Horace is two thousand years old, but it touches even the non-believer in golden ages. How fortunate to live far from the troubles of economic life, far away in a lonely place. Man will find time to think of himself, to dive into his own soul and to salvage the treasures hidden there. He may become a saint or a poet or perhaps a mathematician. Mathematics existed and was appreciated long before there were motors and telegraph lines; it is not a child of the age of engineering though certainly the printing press, libraries and easy communications are greatly helping its advancement; we must be grateful for all these institutions. But the only really essential thing for mathematics is the man and his mental power. In his own soul he finds the material to build up a new world, material as unpretentious and simple as e.g., the numbers and the notion of space.

The idea of number contains "plurality" as well as "order"; this so to speak inborn interconnection of two important notions in one, makes it particularly fit as a basis of further logical developments. The creative will of the thinking mathematician now intervenes and starts operating on the original objects, e.g., he might combine two pluralities into one by addition. By reversing the operation he gets a new operation, the inverse operation; other operations can be created by combining operations by a rule involving a certain order. Similarly the combination of the same operation with itself, the iteration. involves an index of plurality indicating how often the operation is repeated. As pluralities can be expressed by numbers, the iteration of an operation AA can be considered as a new operation which depends on one parameter more than AA. This "promotion of an index of operation to a numerical parameter" is a very important method; it leads to new problems at a later stage when the notion of number is generalised. A new and important step is taken when the operations are made new elements, the "operators"; then the combination of the former operations is considered as an operation on those new operators. In this manner the sphere of entities to be considered and of mental activities to be performed by the mathematician increases indefinitely. Every system created in this way creates an opportunity for a new system formed by the automorphisms of the first one, but the seed of this branch of mathematics is the idea of number combining in itself "plurality" and "order". In this way the original idea of number re-enters mathematics at every stage of investigation. The objects of the operations may be different from numbers; they may even be freely created elements which have no meaning by themselves and there is no limitation for the selection of operations except that they have no self-contradicting properties. However the natural numbers are coming in of necessity as "pluralities" and "indices of order" so that when replaced at one place by generalising notions, they return so to speak at a higher plane. So mathematics shows the basic importance of the concept of numbers in the fabric of our mind; moreover it furnishes also the proper method to investigate the true meaning of "space".

Although geometry has been treated through millennia, it is still an inexhaustible well of new ideas. Not only masses of special investigations are produced every year, but it seems that the problem of space appears under an essentially new aspect to nearly every generation which is active in mathematics. There is no line of demarcation separating geometry from that branch of our science which is based on the notion of number; they penetrate one another and they have grown together like twin-trees which show only a single stem to the observer though they are springing forth from two separate seeds. Space appears to be an ordered system with a type of order fundamentally different from that of the natural numbers. There is no succession of discrete entities in space and no natural starting point; one is dealing here rather with a homogeneous and isotropic medium without gaps. Notions like point, direction and measure help to disentangle this maze. The latter one furnishes the possibility for applying arithmetic. In doing so, one obtains numbers as "proportions" and this gives another, a geometrical, concept of numbers. Thus plurality, order and proportion are three pre-mathematical ideas from which one can start to obtain the notion of number. Mathematics is based on all three of them; the fact that the further development does not lead to logical difficulties which are more serious than those treated in the theory of the continuum, shows that those basic ideas can be considered to be compatible.

These few remarks show already how mathematics discloses the nature of the fundamental elements of our intellect or, at least, of some of them. Simultaneously it amplifies ways of thinking - namely the mathematical ways - by all those methods which form the mathematical literature. An enormous display of intellectual force and dialectic skill, but not a justification of mathematics, unless mathematics is considered as a value by itself without any reference to anything else. If however one adopts this point of view, the question: "Why mathematics" has no sense. On the other hand, we renounced a claim based on the value to be attributed to engineering and similar applications of mathematics. So we got the net result that mathematics is the outcome of some basic ideas of our mind which we feel compelled to investigate and which by the help of progressing mathematical investigation are proved to be interconnected in various ways. One could speak of the introspective value of mathematics. I do not suppose that any other intellectual method has reaped so many and so important results for introspection, but we should not forget or conceal the fact that these results are the harvest of millennia. However the introspective value of mathematics is not its only justification. We should reconsider the case and for this purpose let us ascend again from the depth of the human soul and the seeds of the notions to the surface of life. Using the deductive method I shall shortly review the reasons why individuals and peoples deal with mathematics.

At all times and in every country, there is the "mathematics of the bazaar" the arithmetic of buying and selling, of counting coins, goods and transports. Through ages it has changed very little; only in modern times it was increased by subjects like actuarial science and economic statistics which sometimes apply advanced mathematical methods. Besides this commercial arithmetic, other types of mathematics have been developed in the centres of culture throughout. history; they owe their origin to a great variety of different reasons. For instance the geometry of ancient Egypt is based on the needs of topographic survey. The mathematics of the Middle East is influenced in the first place by problems of astronomy; this holds for the period of the cuneiform characters as well as for the much later period of the Arabic mathematicians. The scholars of ancient India developed a mathematical science in which the theory of numbers prevails; some of their achievements have anticipated results of western scholars living in later centuries. It seems that besides astronomy and chronology, the requirements of the Vedic ritual are responsible for the ways on which the Hindu mathematicians of those days proceeded. The mathematics of Hellas starts from the philosophy of Pythagoras who claimed to have found the leading principle of the world order in the notion of number. On this foundation, the scholars of ancient Greece built up their mathematics which was the prototype of all theoretical sciences. At the present time mathematics is used all over the world for largely differing reasons. Everybody needs the "mathematics of the bazaar", a few are devoted to mathematics for its own sake, others apply it for research on astronomy, physics and other branches of science, a great number of people use it for engineering, navigation, survey, actuarial work, statistics and last not least for education. Let us postpone the last item; it will be discussed separately later. This short review furnishes as net result, that mathematics is mostly practised for some kind of application and not for its own sake. Thus we have to take into account the applicability of mathematics as one of its important features even if we refuse to derive its value from the value or non-value of those applications. It is a strange fact that mathematics whose habitat has been found in the human soul has so many applications in the outer world; it is one of those mysteries which we do not recognise as such because we got accustomed to them. Nevertheless we should try to give some kind of explanation.

Some clue could be found in Galileo's remark that the laws of nature are written in the language of mathematics, but this statement itself is made in the language of a particular philosophy and therefore it must be interpreted first. Laws are human institutions though the underlying principle of justice is of a higher nature. They are imperfect; their value is restricted by geographical and time limits and many offences are committed against them. However the laws given by the Creator to govern nature and supposed to be formulated in the language of mathematics, are in no perceptible connection with the idea of justice; they are eternal and universal; they cannot be broken; they are so perfect that not even room is left for an interference by the Almighty himself. Scientists have established several "laws of nature" and have expressed them in the language of mathematics. So they became able to explain the phenomena of nature and even to predict what will happen on earth and on the firmament whenever certain conditions are satisfied. Some scientists believed that only lack of sufficient mathematics, skill and incomplete knowledge of the physical data concerning the present state of nature prevented them from pre-stating all the future happenings in nature, since these were supposed to be uniquely determined by the laws of nature. However, experience has confirmed those laws of nature found by the scientists only inside a certain sphere of phenomena; they have to be revised to cover a wider sphere and this procedure of alternating extension of our knowledge of physical facts and revision of the assumed laws of nature persists. Thus the laws found by the scientists are only approximations to the true laws of nature - if there exist such laws. The statements of the physicists are made in the language of mathematics and we may suppose that the same will hold true for the statements to be made in future times, but even from this supposition one cannot conclude that the true laws can be announced in a mathematical form. The scientists of the present time are more modest in their aims than those of the 19th century. Neither do they pretend to know the laws given to nature by the Creator, nor do they expect to obtain this knowledge; they only attempt to lift a little bit the curtain which veils the mystery of nature. I suppose that most of the contemporary scientists when speaking of laws of nature, use this term only as a parable and so they do in many other cases. What seem to be fundamental notions of natural science, were known to the philosophers to be categories of human cognition, but the majority of the physicists were hardly conscious of the implications of this statement before the insufficiency of those notions for a complete description of the happenings in nature became obvious. We lived to see that the categories of time, space, causality can be applied to physics with limitations only and many other notions have been disclosed to be mere parables and images which are very helpful when used inside certain boundaries, but become useless outside of them. Theoretical science has reached a state where all the notions have become questionable and the mathematical method alone has remained permanent.

Thus it appears that in the human soul there is nothing so appropriate to help us for an understanding of the occurrences in the outer world as mathematics; it forms an important link between ourselves and the world around us. It would be a daring attempt to value the treasures of our soul and to put them into an order of merit, but certainly mathematics would not stand in the last place since it helps us to know something of the inner as well as of the outer world. This proves the importance of mathematics in the system of our knowledge and it shows that this science serves other purposes than the private enjoyment of a group of people, called mathematicians, who are obsessed by a cranky desire to investigate topics whose knowledge is of nobody's concern.

If one considers the high importance of mathematics, thus explained, one will understand why Plato did not admit anybody to his Academy who had not a fair knowledge of geometry. From that time dates the close interconnection between mathematics and education. It seems to be necessary to add a few words about this item because whenever the question "Why Mathematics?" is asked in public discussions, it concerns mathematics as a part of education, in particular of general higher education. That elementary mathematics is necessary for the requirements of daily life, is obvious, nor is there any serious objection against the training of experts in that subject by special courses. The debatable topic is whether some knowledge of mathematics above the level of elementary arithmetic is necessary for a well educated man. We should approach this question without any professional bias. Even Felix Klein has admitted that there are very talented persons who are incapable of following the ways of mathematical thinking and that they had better leave mathematics alone. One might however argue that just these people need more than others the strict discipline imposed by mathematical logic. One may expect that the rigorous form of mathematical conclusions will sharpen the critical sense and stiffen the student's mental self discipline; moreover his experience that he is able to check the correctness of mathematical investigations himself by logical considerations alone is likely to enhance his self confidence and the feeling of responsibility for his own thoughts. Dealing with mathematics is bound to kindle the flame of love for harmony, truth and spiritual beauty and to give strength to the intellect. If these considerations are correct, one must expect the professional mathematicians to be rare specimens of human perfection since they are living all their life under the purifying influence of their science. However, looking at the biographies of eminent mathematicians - e.g., perusing Bell's "Men of Mathematics" - one will find that even this class of people are not free from short-comings of character. In the glaring light of biographical investigations some unpleasant features become visible which fortunately remain hidden to most of the contemporaries. Thus there is no wonder that my own impression about mathematicians is much more favourable. I like to remember the harmony and the spirit of good fellowship prevailing at the meetings of this Society at Lucknow, Hyderabad and Aligarh and the kindness shown by distinguished mathematicians of this country to me as a newcomer; moreover I feel much indebted to many mathematicians of other countries - living or deceased - to my teachers and to those with whom I used to exchange ideas. My general impression is that mathematicians, compared with other individuals of the same social classes, are amenable people, a rule which - as it has to be - is confirmed by very distinctive exceptions. However I have some doubts whether a few periods a week of mathematical teaching has a favourable influence on the character of the average student. As a matter of fact, educationists are pointing more to the training of intelligence by mathematics than to its influence on the character. So the controversy concerns the question: How far is mathematics helpful or necessary for the training of the intellect and how far should it be replaced by other means of instruction? A new element is coming into these discussions through the claims of several professions for a better and more specific mathematical training of their recruits and the suggestion that the foundations thereof should be laid by the general education. The complaints of the engineers are well known. Moreover I may mention that in public discussions arranged by the Calcutta Mathematical Society, a very distinguished American hygienist, strongly supported by two of his Indian colleagues, urged the necessity of some training in statistics for all the members of the medical profession. It is very probable that the same need will be felt soon for various other practical activities, e.g., general administration. To satisfy such claims, important changes in the system of teaching are necessary, but these are meeting many difficulties. Unfortunately in many countries the curricula for teaching mathematics are still based more on tradition than on practical and educational considerations. Besides this outer difficulty, there is another one originating from the very nature of mathematics. It is impossible to form a curriculum simply by putting together those topics which are of particular practical or educational value. In mathematics everything is interdependent and every theory needs its proper foundation. Therefore a good syllabus must be a compromise considering many endeavours and necessities; in every country every generation of mathematicians and educationists has to fulfil this task without being afraid of its difficulty. In mathematics first things must come first, whereas education wants a progress from the easy to the difficult; these two claims are often conflicting. For this reason the teaching of mathematics has its particular difficulties and it demands the highest standard of knowledge and educational skill from the teacher. Sceptics may ask whether the results justify such efforts: I personally believe that in every country better results could justify even much higher efforts. A detailed treatment of these educational questions as far as India is concerned must be left to the discussions during this meeting. Also at previous meetings, our Society has spent a considerable time on problems of mathematical education. These concern mathematics for education as well as education for mathematics. As mathematics has been proved to be an important branch of human activity, mathematicians have the duty to hand down their knowledge to succeeding generations and to make suitable educational arrangements for this purpose.

With these remarks on education, I may conclude my observations why mathematics is practised and why it must be practised. It has often been discussed whether it is an arts or a science subject; to many people it seems to stand on the border-line. I should not like to put it in this way. While arts-subjects are those concerned with the inner, and science-subjects those concerned with the outer world, mathematics is an arts-subject and simultaneously the basis of all the science-subjects with which it is connected in an indissoluble manner. So it stands as a symbol for the unity of the human mind.

Last Updated July 2026