Public Perception of Mathematics

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)
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.
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.
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.”
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 sub graph).
We conclude: 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 sub disciplines 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.
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