R Vaidyanathaswamy's Presidential address

R Vaidyanathaswamy delivered the Presidential Address to the Twelfth Conference of the Indian Mathematical Society. The Conference was held at Aligarh at the invitation of the Muslim University, 27-30 December 1941. The Address, delivered at the opening session on 27 December, was published in The Mathematics Student 10 (1) (1942), 12-17. We give a version of this address below.

R Vaidyanathaswamy's Presidential address

Mr Vice-Chancellor, Ladies and Gentlemen:

It is with great pleasure that I acknowledge the warmth of the welcome which has been accorded to this Mathematical Conference. When I look back on the previous conferences of the Indian Mathematical Society, it seems to me that the present conference will prove to be memorable in many ways. In the first place, in contrast to the peaceful atmosphere of our Mathematical deliberations in the past, we are meeting for the first time under the threat of impending war-conditions. The association between the tense exigency of the moment and the calm detachment of scientific thought is not perhaps so unnatural as it may appear at first sight; our temporal consciousness moves always between the two poles of the moment and the eternal, but these two poles even in their opposition seem to be linked in a mysterious kinship; for how else can we explain the fact that often in moments of intense stress, our minds get a sudden access to sensibility and becoming as it were, released from a prison, rise to a perception of universal things? We may recall that Archimedes, who with Newton and Gauss is reckoned among the three greatest Mathematicians of the world, was in a state of concentration on a mathematical problem when the invaders of his city broke in upon him. Again, this conference is the first held by the Indian Mathematical Society in collaboration with a sister Mathematical Society; and this co-operation for a common cause is a happy augury for the future advancement of Mathematical research in this country. And lastly, this is the second time in succession that our Conference has been welcomed by a progressive University devoted to Muslim Culture - a culture that has made a respectable contribution to the early development of our Science.

The fact that in the past our Conferences have been enthusiastically received and welcomed by various Universities shows that, though Indian Universities are mainly engaged in the dissemination of knowledge, they are not blind to the fact that the primary function and justification of the University is research or the creation of new knowledge. It appears to me that the Universities in India are in a stage of transition at the present day. On the one hand, they are fettered by their past record of a mechanical reception of Western Knowledge and its mechanical transmission. Now, when the ideal of vernacularisation of all teaching is in the air, they are showing signs of discomfort and are beginning to be seized by stirrings towards a more creative and a fuller intellectual life. While we look forward to a post-war India with a revised political status, it will not I think be any undue optimism, to hope that Higher Education in this country will come to its own, and the Indian Universities will begin to function as the centres of higher knowledge and research and intellectual leadership in the arts and sciences.

It will certainly prepare the way for this inevitable transformation, if we realise that college instruction cannot be separated from research except to its detriment and degradation. The syllabus in Mathematics for a degree course, for instance, consists of a body of doctrines, together with the training for manipulating them in solving certain types of questions; these doctrines are advanced in text-books and by teachers, as if they had no background and no connection with any large human interests; as if they had always existed in their own right, because they were absolutely true and final - with that species of brutish finality which popular opinion finds it easier to attribute to Mathematics, than to sciences which are partly or wholly experimental. But this is a totally distorted view of the matter. The doctrines in our Mathematical syllabus represents the net gains of some two millenniums of intellectual effort of the human race; each particular idea or relationship was first manifested to an individual human intelligence after a long travail of seeking and the whole body of doctrine called Mathematics has been gathered and handed down as a precious relic, because of its inestimable value and significance for those ideals and for that fulfilment, for which the human race is striving through the ages. If this outlook towards Mathematical truth cannot be instinctively communicated by the teacher to his pupils, then what the teacher performs is not the teaching of Mathematics but its mechanisation and vulgarisation. And as a rule, the teacher will not be able to communicate this kind of feeling for Mathematics, unless he has himself gone through the great experience of Mathematical discovery. To the Seeker after Truth, this cardinal experience of discovery is a turning point, and may well be termed a baptismal experience, as it marks his birth into a world of deeper reality - the world of ideas. The experience of discovery is roughly the glimpsing of a new idea, by which the past knowledge is re-ordered and re-oriented, so as to exhibit hidden relationships and new and unsuspected values, so that there results a thorough transformation of outlook. A prosaic description of this kind does not really convey the sense and momentousness of this cardinal experience of the enlargement of human consciousness; for that, one must hark back to the Vedic and Puranic Symbolism. The Ushas of the Rig Veda, the Deity of the Dawn is nothing else but this experience, and the glamour of this Deity, and the number of the hymns addressed to her and their fervid passion, show how deeply the Vedic Rishis had realised and experienced the recurrent enlargements of the aspiring evolving human consciousness. Similarly the glamorous figure of Lakshmi rising from the ocean of milk churned by the Devas and Asuras is the pre-eminent symbol of something absolutely new and original, something fresh from the primordial mint of creation, rising in the human consciousness and miraculously transforming the whole outlook and revealing a new world of values. It is only that person who has experienced the great cosmic mystery of the rising of the original Lakshmi from the churned Ocean of Milk that can penetrate behind the Symbolism of the cheap vulgarisations of the figure of Lakshmi, which can be bought in the bazaars; in a similar manner, it is only the teachers who have gone through the baptismal experience of Mathematical discovery, that can convey to his pupils the inner significance of the text-book Mathematical doctrines, relive with them the original glamour of their first discovery, their first revelation to a thrilled human mind, and communicate a feeling of their place and value in the journey of the human race, the Epic of Civilisation.

The Indian student or Indian Teacher of the present day wishing to undertake Mathematical research, is confronted with the vast Mathematical material which is the legacy of the 19th century. The 19th century has been well called the Golden Age of Mathematics, its achievements belittling that of all previous centuries taken together. The developments of this century in "Analysis" and "Function-Theory", in Algebra and Geometry, in Arithmetic and in Invariant-Theory, in Mechanics, have so changed the face of Mathematics as to alter it beyond recognition and have won for it a far-flung dominion. Indeed some of the things which are a result of the work of this century, and which we teach young pupils of the College classes belong to a conceptual level that would baffle the understanding of the mighty intellects of antiquity or of the Middle Ages. The basic ideas of the 19th century work do form or should form part of the course in our Honours and post-graduate classes. The Mathematical aspirant who wishes to push on to research, should of course discipline himself thoroughly in the methods of great masters of the 19th century; this would not be difficult to accomplish in any particular topic, though the extent and refractory character of the material would make it difficult to do in several topics at once. The worker would then be naturally led on to extend the scope of these methods or to complete the solution of some problem by taking it up at the point where it had been left off. This would be the usual way which workers might be expected to take and do actually take both here and in other countries. But is this the right way? I think not. Let me explain why. Some young Indians inspired and fascinated by Euclid's Elements, are enticed to further research and do succeed in finding ingenious proofs of difficult riders or in constructing elegant theories within the ambit of Euclid's method and view-point. Others, of a more sceptical or valorous disposition attempt a proof of the parallel postulate by the classical methods of Euclid; these are by no means cranks or irrational persons; for if you limit yourself to the view point of Euclid, it is not at all certain that the parallel postulate could not be proved. Euclid, of course, guessed that it could not be, but it was only a guess, and he might have well been wrong. It was only in later times and from a wider point of view that it became possible to prove that the parallel postulate cannot be proved. My point is that in neither of these cases is there any really creative work, because there is no advance beyond the point of view of Euclid - the point of view of two millenniums ago. On the other hand what the true nature of creative work in geometry should be is well demonstrated by the actual historical course of development. The transformation of Euclidean Geometry into Geometry as we think of it today was effected by the impact of some half-a-dozen new ideas in succession on the material of Euclidean Geometry. Each of these impacts irradiated the material and raised it to a higher plane of value and significance. The Treatise on the Conic Sections of the Greek Geometer Apollonius was not one of these impacts, because though an extension and addition to Euclid's work it was not a creative addition in the true sense, it belonged to the same value-level and the same significance-level as Euclid's elements. These impacts may be easily enumerated; the first was Descartes' discovery of Algebraic Geometry, the second was the principle of continuity associated therewith, which brought in imaginary elements into geometry and the concept of n-dimensional geometry. The next was the discovery of non-Euclidean Geometry with the conviction of its reality. The next was the discovery of projective geometry with the importing of infinite elements. The next was the concept of group and invariant culminating in Klein's grand generalisation of geometry as the invariant theory of a transformation group. The last was Gauss' discovery of differential geometry and Riemann's concept of curvature of a manifold. Each of these new ideas illuminates the subject matter in such a way that it might well be called a "DAWN" in accordance with the Vedic imagery; and this name is all the more appropriate because the illumination in each case has come in response to a seeking for light in order to clear up some felt darkness in the subject. Thus in brief Euclid's conception of geometry and ours are separated by some half a dozen dawns.

This being the nature of creative research, the present day Mathematical aspirant should seriously ask himself whether in dealing with the rich legacy of the 19th century, he is really bound to limit himself to its view point and method, whether there have not happened subsequent dawns, dawns of this century, the illumination of which has yet to play on the older material transforming it and raising it to its own level of value and significance. If there have been such dawns, the path of creative research lies most definitely in the fulfilment of the Time-Spirit by assimilating the older material to the newer light.

We have today arrived at a deeper insight into the nature of Mathematical thought than was ever possible before. As the culmination of the researches in the analysis of mathematical reasoning initiated in the last century, we realise today that Mathematics should be conceived as 'the class of all deductive systems' or as 'the analysis of structure'. This is no doubt a highly abstract view, but abstraction where Mathematics is concerned, is not a flight from the concrete, but a temporary stepping back from the subject matter, in order to get it into clearer perspective and return upon it with greater power and insight. Looking back on the past, we may recognise that this intuition of the nature of Mathematics has been always present from the earliest times as far back as Euclid, though in a germinal form without rising to self-awareness.

Another curious feature of this modern view of Mathematics is the laying bare of the peculiar relation which holds between Mathematics and Logic - a relation, not of one-sided dependence as was thought, but of two competitors in a race each of whom tries to outstrip the other. For, it turns out that logic - or rather, any particular level or system of logic - has a structure of its own and can be formalised and presented as a deductive system, in which the processes and reasonings depending on the actual 'content' as opposed to the 'form' of the system belong to a deeper unformalised level of 'meta-logic.' This must necessarily turn out to be the case with a human mentality capable of deeper and ever deeper levels of reflection on its own processes.

The 'Structures' or 'Deductive Systems' of Mathematics may be of a very general or even of an arbitrary character. But in practice, the purposive character of Mathematical science brings in some limitations. Historically Mathematics had the task of elucidating the problems of practical life. Its central and distinctive contribution in this regard is the concept of the 'real number,' by which the sense world can be 'measured,' and its manifold relations explained. This concept runs like a spinal cord through the whole of Mathematics, furnishing a solid support to its abstract structures and a centre of reference to its boundless peregrinations. Indeed, when any general Mathematical theory or result has to be brought home by an illustration or to be used or applied, we have to bring in the real number directly or indirectly. It will not be unnatural then to except the typical Mathematical Structures to have a close bearing on the real number concept, to hover round it, as it were, or even to have been implicitly involved in its development.

The concept of the 'real number' has developed through four stages, the cardinal number, the natural number, the rational (or fractional number), and the real number, and we do find Mathematical Structures typical of each of these stages of its evolution. The 'Cardinal number' is the signless integer used in counting, and the addition of these cardinal numbers must be explained in terms of the union of sets without common elements. Thus the notion of cardinal number must be logically explained in terms of set-algebra. But the notion of set has a definite content, and is therefore 'concrete' and not 'abstract' according to the standards set by Mathematics. To get at the formal characteristic of set-algebra, we observe that both set-union and set-intersection can be explained in terms of the relation of set-inclusion. Therefore the abstract Mathematical Structure revealed in set-algebra and in the notion of cardinal number is the 'partially ordered set,' namely a set of elements constituting the field of a binary-reflexive transitive relation, formally analogous to set-inclusion. It is not surprising that the structural idea of the 'partially ordered set' includes logic also among its applications, in view of its close connection with the fundamental logical notion of 'set.'

The cardinal numbers are not closed for subtraction, and the impulse which conceives the 'natural numbers' or the set of positive and negative integers, is clearly the impulse of 'completing the group.' Thus the Structural idea which is manifested at the second stage of evolution is the idea of 'group.' The passage to the rational number is similarly effected by completing the multiplicative group of the natural numbers other than zero; this brings forth the structural idea of 'field' which is or rather ought to be, the subject-matter of Algebra. In the final stage, the notion of real number is arrived at from the rational number by a coup d'état accepting an infinite process. According to Dedekind, the motivation in this process is to arrive at an analysis and a logical explanation of the geometrical intuition of 'Continuity.' The Mathematical discipline which analyses and investigates the structure of the notion of continuity is called 'Topology.' Accordingly the structural moment revealed in the passage from the rational to the real number must be called the 'topological moment.' Thus corresponding to the four stages of evolution of the real number, we have four principal structure-types, the 'partially ordered set' and 'topology' at the extremes, and the intervening structural levels of 'group' and 'field' which are usually included under 'Algebra.' These are of course 'elementary' structure forms, and may be found mixed up together in the background of any actual piece of mathematical reasoning. In order to study these structures effectively, Mathematics would also have to deal with generalised forms of a looser structure, as for instance 'Semi-group' or 'ring.' Also, an infinite variety of secondary or derived structure-types may arise, based on these; as for instance in Geometry, where we have to deal with the structure of sets of linear equations in a field. Thus, in fine, we have to ascribe a central position in Mathematics, to the four structural moments revealed in the evolution of the real number, without prejudice to the fact that Mathematics is concerned with deductive theories based on any system of postulates whatever.

Every discovery of new and far-reaching ideas must be followed up by a period in which their scope and applicability to the existing material is surveyed and worked out. Such work is necessary not only for an increased insight into the existing body of knowledge, but also for giving content to the new ideas and fixing their bearings. It is work in this direction that is called for at the present moment, and I feel strongly that Indian Mathematicians and workers should orient themselves accordingly, so as not to be behindhand in doing their share.

Last Updated July 2026