During innumerable events, interactions and meetings held from time to time with various sections of the society comprising most prominently those who come from the educated segment of the society including especially the teachers, doctors, engineers, lawyers or journalists, the question that invariably gets asked of me is something like this: “What on earth does research in mathematics entail when the fact remains that ‘almost everything that needs to be known about mathematics is already there, and so very well known?’ ” Of course, their reference to mathematics is solely in terms what they have learnt in the school: adding, subtracting, multiplying or dividing numbers and that with the advent of computers and other fast computing devices, the speed of carrying out such manipulations involving large numbers has already reached a crescendo. Such an ill-informed perception of mathematics in the public eye is perhaps one of the main reasons why mathematics does not figure as a career option among the students when they have to make a choice immediately after passing the 12th grade.
From the standpoint of a well-bred mathematician who has engaged himself in teaching and doing research in mathematics throughout his professional life, the question may indeed come across as inane and insipid. However, on my part, I have come to realise that the question merits a response even as it betokens an absence of a certain mathematical culture where the need becomes still greater to address the reasons for such a poor perception of mathematics in the society which is seen merely as a tool to manipulate columns of numbers. And those of them who are willing to take a more ‘informed’ and generous view of mathematics would go as far as to concede that mathematics acts as a useful tool in sciences, but rarely a trifle more than that!
At the outset, I must confess that it’s not easy to carry conviction to the sceptic even as their ignorance about the issue under discussion may be pinned down to the absence of an ethos where mathematics was rooted as an important part of its culture and where people learnt of mathematics as an endeavour that made sense beyond the drudgery of adding and multiplying numbers. The point is that the best way to see mathematics as a highly rewarding human endeavour is to be one with it, if you like, in order to enjoy the act of being involved in it regardless of whether it’s pursued for its own sake or employed towards a better understanding of the world around us. That brings up two most important aspects of mathematics- the abstract part of it which is pursued for its own sake and that part that is used as an important tool in the service of humanity. These two sides of the picture are beautifully portrayed in the following immortal words of two of the greatest minds of the twentieth century:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. (BERTRAND RUSSELL).
“One of the remarkable things about the behaviour of the world is how it seems to be grounded in mathematics to a quite extraordinary degree of accuracy. The more we understand about the physical world and the deeper we probe into the laws of nature, the more it seems as though the physical world almost evaporates and we are left only with mathematics”. (Roger Penrose)
Power of abstract thinking/theory
Between the two extremes involving the uncompromising allegiance to rigour coupled with the importance of a mathematically correct and logical proof of mathematical truth on the one side and the ‘use and throw’ sort of approach of scientists and engineers towards mathematics, there arises the utmost need for the physicist to adopt what may be called the ‘middle path approach’ towards mathematics. That would entail a conscious effort on their part to have a somewhat deeper insight into the part of mathematics that they have used in their researches.
While obviously being useful to the physicist in their understanding of physics under investigation, such an approach would also allow them to have a better appreciation of mathematics that has been used in their understanding of physics. We are told how recent work in string theory has led to spectacular insights into certain parts of mathematics which had hitherto remained a mystery from a purely mathematical standpoint. Come to think of it, fundamental research creates the intellectual climate in which our modern civilization flourishes. As Enrico Bombieri had famously said: “Knowledge even when it is not motivated by short-term goals is always very precious”.
Without intending to overwhelm the reader with the technical terms that follow, the idea is to stress the point that the effort to understand the structure of abstract objects such as the Monster group, the Jones polynomial or Donaldson’s exotic differentiable structures on the four-dimensional Euclidean space which was inspired purely by considerations of the search for beauty and truth surrounding these abstract objects has led to the totally unexpected realisation that these theories are in effect different aspects of quantum field theory in dimensions two, three and four, respectively. It is remarkable that something existing in the human mind as pure abstract thought should have such a significant role to play in our understanding of the universe around us. Here, it is pertinent to mention what C. J. Jacobi had famously said about Joseph Fourier, the originator of Fourier analysis:
“It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honour of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world.”
Mathematics in real-life situations
Whereas the raison d’etre of mathematics continues to remain the platonic pursuit of (mathematical) beauty for its own sake, it’s curious to note that in the applications of mathematics outside of it, there’s a surprising correspondence between the level and depth of a mathematical idea and the significance and impact of its applications in a given real-life situation. The point is that the part of mathematics that finds use in science and engineering happens to belong to the genre of mathematics which is generally considered to be accessible, relatively simple and easier to grasp, of course not for the uninitiated.
We illustrate this by referring to the role of elementary arithmetic in counting and the manipulation of numbers in our everyday life. On the one hand, the use of elementary number theory in the ATM-credit card transactions appeals to no more than the application of an elementary fact of number theory, the Fermat’s little theorem. On the other hand, it is a bit of a stretch to expect the so-called Fermat’s last theorem whose proof is known to be too deep and way too hard to lend itself to applications in real life situations, at least in the foreseeable future.
Let us remember that there is no such thing as a free lunch. To suggest that useful applications of mathematics could be conceived without spending time and effort to pursue and develop pure mathematics is akin to hoping that you could get the tourism benefits of such hitherto uncharted spots as, say the Bungus valley in Kashmir without caring to invest in exploring it, which in any case was already there to be explored. The Voyager space mission is yet another example involving huge investment of money by NASA to explore outer space that has led to incredible breakthroughs in deep space communication that allowed NASA to transmit pictures and other data from the Voyager space craft to earth.
It is interesting to note that apart from the sheer curiosity to explore and know better about the outer space that entailed prodigious investment of money in the project, the key component of the exploration was provided by deep researches in the important domain of pure mathematics involving the abstract phenomenon of ‘sphere packing’ where the Ukranian mathematician Maryna Viazovska’s breakthrough work in dimension 8 won her the Fields Medal in 2022. On the same analogy, it is inconceivable that the mathematicians studying hyperbolic manifolds would have suspected the application of these ideas to quantum error-correcting codes, had it not been for deep thought and reflection underpinning its exploration!
Mathematics in Social Sciences
With the increasing use of mathematics in social sciences, the thinking is gaining ground that “mathematics will receive as much of its direction and vigour in the future from problems in social sciences as it has received in the past from the physical sciences”. The use of basic sampling theory in drawing useful inferences from the exit polls regarding the final outcome of the polls or in the approval of a vaccine, or in using basic graph theory in providing public utilities/services to the customers are some of the commonplace examples of this interface.
The last application involves the need to provide public utilities to, say three households each of which is required to be connected by cables to the centres of three companies which supply electricity, telephone and internet, respectively such that the cables are laid underground in such a way that they don’t cross over each other. It turns out that howsoever hard one may try, it is impossible for the companies to find a layout of the cables so that the utilities are provided directly to the houses without being routed through the other houses. The fact that the connections cannot be thus ensured to be provided follows as a consequence of an important theorem in graph theory asserting that the graph G which is used to model the above problem is not planar! (In technical terms, this amounts to saying that the graph G contains the (bipartite) graph as a subgraph).
We conclude our discussion by harking back to the question that was posed in the beginning of this article: what does mathematical research entail after all! The broad contours of what actually occupies a research mathematician involve the study of mathematical structures and a search for order, symmetries, and patterns hidden inside the structures and alongside that, the possible existence of a superstructure within which the given structure is subsumed as a substructure. That kind of an approach broadly belongs in the domain of “Theory Building” where one looks at the ‘big picture’ and in the process, happens to discover unexpected connections with other disparate subdisciplines of mathematics.
Here one comes across the grand unity of mathematics connecting ideas from apparently unrelated areas of mathematics – a phenomenon that’s unique to mathematics. The famous Langlands program as a ‘grand unification theory’ in mathematics proposed by Robert Langlands seeks to connect such disparate areas of mathematics as number theory, algebra, analysis, geometry, representation theory and many other disciplines into a single unified theory. That, of course, constitutes the apotheosis of the search for truth and beauty. Surprisingly, it is the platonic search for truth, beauty and order that almost always opens up avenues for the application of abstract ideas across diverse spheres of human activity involving science, technology, humanities, art and culture and scores of sundry human endeavours.
Prof. M. A. Sofi, Professor Emeritus,
KU, Srinagar
