What is Mathematics?

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Known as the queen of sciences, Mathematics, perhaps unlike other disciplines has defied definitions. It is the language of science. This book introduces mathematics, as the study of logical structure of patterns and focuses on its distinct aspects like deduction, representation, transformation, beauty and meaning. Using quotidian approaches and stories as metaphors, the book not only explains various mathematical concepts but also its history as well as India's contribution. Written in a lucid style, the book addresses the needs of students of mathematics, teachers, researchers as well those interested in the subject.

Balkrishna Shetty studied Mathematics at Presidency College, Kolkata and IIT, Kanpur. After a year at the School of Mathematics, Tata Institutes of Fundamental Research, Mumbai he spent over two years in the Indian Statistical Service. Shetty joined the Indian Foreign Service in 1976 and worked in Indian missions in Geneva, Dhaka, Moscow, Singapore and Paris before serving as Ambassador of India in nine countries, including Senegal, Bahrain and Sweden. A keen scholar of mathematics, he participated in the International Congress of Mathematics, Hyderabad, India.

Mathematics has always been an integral but subtle part of our knowledge and culture. However, the last five hundred years of our history have confirmed Mathematics as among the most treasured intellectual products of mankind's civilisation. It has not only provided deep insights into understanding the world around us but has also made it possible to harness the forces and resources of nature to alter and improve spectacularly the material aspects of our individual lives and thereby radically transform the collective destiny of the human race. Broadly speaking, the Agricultural Revolution can be said to have had links with geometry and numbers ; the Industrial Revolution was forged in the intellectual fire fed by the fuel of the mathematics of Calculus; and today's Information Revolution is being driven by the abstract mathematics of axiomatic theories.

The past five centuries have also shown that the difficult and abstract mathematics of one century can become the staple diet of scientific theories and the concomitant technological developments of the next. The applications of Mathematics, and even of its advanced techniques, earlier confined mainly to natural sciences and engineering, have now spread to economics, other social sciences and even to day-to-day practical businesses. This has been facilitated by the widespread applications of statistical methods and the use of the computer, the latter itself being a product of mathematical developments in the first half of the twentieth century. Thus, increasingly sophisticated optimisation methods now drive our business processes, while abstract and advanced mathematical methods help in medical diagnosis, undergird security of banking operations, and ensure space-flight control, to mention only a few instances out of literally thousands. As mathematical research reaches out to complicated non-linear systems, like neurological networks, biological organisms, social groups and political structures, Mathematics can be expected to play a much greater and more critical role, directly and indirectly, in our daily lives.

However, the broad impression about Mathematics among a significant proportion of the general public still seems to be that the discipline is mainly concerned with convoluted calculations with numbers, meaningless manipulations of symbols and confounding compilations of propositions. It is also not uncommon to come across people who believe that all that needs to be proved in Mathematics has already been proved! This is perhaps due to teaching Mathematics the way Hindi was once very superficially taught to those who knew only English, whereby they learnt English sentences which sounded like the required Hindi phrases, such as, `There was a cold day' for Darwaza khol dey' (i.e., open the door). Needless to say, while such shallow 'translation' may open doors of houses, such teaching methods are unlikely to open the doors of the mind to understanding Mathematics! Such methods are all the more surprising as, interestingly and quite contrary to popular misconceptions, Mathematics and the mathematical way of thinking have always been an integral part of our daily lives and deeply embedded in our cultures. They are consequences of the fact that mathematical thinking is very closely related to critical thinking in particular and cognition in general. Indeed, like Moliere's Monsieur Jourdain and the latter's prose, we have been continually practicing Mathematics, very often without being aware of it.

But what is Mathematics? This question naturally comprises a whole host of subsidiary questions, some of the more natural ones being the following: What is the source of Mathematics? What are mathematical objects? Are they discovered or invented? Is it possible to know the nature of mathematical objects? Where do abstract integers and perfect circles exist? More generally, what do we mean by the existence of mathematical objects? How do we discover their properties? Why does Mathematics seem obsessed with logical nit-picking, requiring even self-evident results to be proved? What does a proof really mean? Why is Mathematics filled with symbols? Why are form and methodical approach so important in Mathematics? Why are algebraic manipulations so ubiquitous in the discipline? 'What are infinitesimals and infinity? What is the difference between ordinary points and lines and the 'points at infinity' and 'lines at infinity'? Why has Mathematics shown a tendency to become more and more abstract? Why is it often easier to prove a more general result? What makes a mathematical proposition or its proof beautiful? Why is it possible for the same theorem to be proved in so many different ways? Are mathematical propositions absolute truths? Why is Mathematics so effective in natural sciences?

It is evidently not possible to answer such questions about the nature of Mathematics in an adequate manner by merely pointing to its elementary sub-divisions like arithmetic, algebra, and geometry, or even to its more advanced branches like analytic number theory, modern abstract algebra and general topology, and exhibit some propositions and methods in each of these areas. It seems unlikely that merely solving mathematical problems would give a proper appreciation of the nature of Mathematics. As such, the focus of this book is neither on individual domains of Mathematics nor on solving problems-there are no explicit exercises required to be done or techniques to be learnt. It, instead, provides a simple introduction to understanding Mathematics in a broad sense from the perspective that Mathematics is, at the most basic level, the study of logical structure of patterns. This explains immediately why the focus in Mathematics is on general logical relations and isomorphisms, rather than on the nature of a particular carrier of the pattern; why the discipline is so useful to other domains; why it has benefitted from its practical applications; and why beauty is an integral part of the discipline.

This fresh approach emphasises the more fundamental aspects of Mathematics which are common to all its sub-divisions and branches, by examining the main motifs of the discipline. Some of these motifs are-assumption as the starting point of concepts, theoiems and approaches; set, as the common basic objects of the discipline along with its associated language; and deduction, to discover and understand the logical links amongst mathematical patterns. Other motifs include representation of mathematical object for facilitating comprehension; transformation as a mathematical kaleidoscope for obtaining a better picture of, or a more interesting insight into, a pattern; the methodical approach of the subject, so necessary for avoiding confusion in the midst of complexity and the need for strict rigour; parsimony, as manifest in the familiar mathematical features of analysis, abstraction, and axiomatization, reflecting the mathematical drive to capture compactly the diverse patterns through basic objects and higher-level similarities; its infinite degree of precision, readily visible in geometric figures and infinite series; beauty, seen in its amazing regularities and elegant proofs; and, above all, meaning, for Mathematics must make sence.

Mathematics evokes two extreme kinds of response among lay people; there are a large number who are very uncomfortable with it (and that would date back to their school days) even as they hold the subject in some awe; and there are the others who were very good at it in school but did not pursue it for various reasons. But few in either category seem to have a clear idea on what mathematics is all about. It is not uncommon to come across people who think that calculating prodigies (like Sakuntala Devi) are mathematicians which they are not; I have also heard people wonder what is left in mathematics to discover.

This book attempts to explain to lay people what mathematics is all about; and handles this extremely difficult task very well. The author Balkrishna Shetty, as a young man, wanted to become a mathematician. He was in fact a very promising graduate-student when he opted for a career in Government service. He has however retained some contact with mathematics over the years and his romantic attachment to mathematics seems to have remained intact. I offer him my congratulations on this effort.

I do hope that the book will have a wide readership and help increase informed awareness of mathematics in our society.

Among all branches of natural sciences, it is most certainly Mathematics that evokes the most trepidation. Mathematicians themselves find it hard to communicate to non-mathematicians and hence there is hardly any mathematician who is recognizable, say, like an Einstein or a Feynman. The discomfort or uneasiness which a professional mathematician experiences in communicating to the uninitiated germinates from the fact that Mathematics has a language of its own. In some ways, it is perhaps similar to Law, in the sense that a misplaced comma in an Act can potentially change the context, or even worse, bring in ambiguity. The mathematician faces a similar dilemma when he speaks to non-mathematicians. Should he only speak of the trivial cases and hence avoid any misrepresentation? If so, isn't he still cheating in the sense that he is not giving the right picture? This perpetual balancing and counter-balancing, between what to say and what not to say, is what makes it nightmarish for a mathematician to give a public talk. And this, perhaps, explains the communication gap between Mathematics and other branches of social as well as natural sciences. The present book is an attempt to convey some basic principles of Mathematics in English, avoiding Mathematical English. The author has tried his best to follow the advice, supposedly given to Stephen Hawking by his publisher, that every equation would lose him X-thousand readers. Luminaries like Cauchy, Abel, Gauss, Riemann, etc. appear with brief descriptions about their works. The book offers nice historical snippets about these mathematicians as well as their mathematics, which is accessible and gently paced. The chapters are interspersed with short anecdotal stories in resonance with the topics discussed. In fact, using various well-known stories as metaphors, the author attempts and succeeds in providing insights into both mathematical concepts as well as mathematical methods. At the same time, on a more serious note, his approach to the definition of Mathematics and its essential themes via these metaphors is interesting and offers some food for thought even to the practicing mathematician.

The book does not demand a lot in respect of mathematical background, and anybody familiar with basic high-school mathematics should be able to go through large parts of the book. At the same time, the book has something to offer to the more advanced reader. For instance, the author describes the famous Ramanujan conjecture on the bounds for the so-called Tau function. Deligne showed that this Ramanujan conjecture is a consequence of the Weil conjectures. The Weil conjectures are one of the central problems in Algebraic Geometry as well as Number theory and was established by Deligne himself. This was one of the outstanding mathematical achievements of the second half of the 20th century and for which Deligne was awarded the Fields Medal in 1978, the mathematical equivalent of the Nobel Prize.

There is a dearth of good mathematics teachers in our country. It is my opinion that the present book should help the teachers to motivate themselves as well as introducing them to "Mathematical culture". Only then can they generate a sense of excitement among their students, reinforcing the belief that Mathematics is an exciting living science.

**Contents and Sample Pages**