K C Chakrabarty: The magical world of mathematics - the charm,
challenges and career prospects
Address by Dr K C Chakrabarty, Deputy Governor of the Reserve Bank of India, at
"Mathemight", a conference organized by the Department of Mathematics of the Vivekanand
Education Society's College of Arts, Science & Commerce, Mumbai, 18 January 2013.
* * *
Assistance provided by Shri A.B. Chakraborty, Shri Sanjoy Bose and Shri Shailendra Trivedi in preparation of this
address is gratefully acknowledged.
Introduction
1. Dr. Rajeeva Karandikar, Director, Chennai Mathematical Institute; Dr. Mrs. J.K.
Phadnis, Principal of the VES College of Arts, Science and Commerce; Prof. Amiya Kumar
Pani, Chair Professor, Department of Mathematics, IIT Bombay; Mrs. Dipta Dasgupta,
Convener of this conference aptly titled "Mathemight", faculty members; other distinguished
speakers and student participants; ladies and gentlemen. I am delighted to be here today in
the midst of eminent scholars and students of mathematics. The Vivekananda Education
Society has successfully completed fifty years in its "Pursuit of Excellence in Higher
Education" and I congratulate the Society on this achievement. I understand that the Society
is organizing frontline seminars and conferences for our younger generation to sow the
seeds of curiosity and to enhance their knowledge base. You may recall that this year, we
are celebrating the 150th birth anniversary of Swami Vivekananda, one of the greatest
spiritual leaders of India. He once observed: "you know how many sciences had their origin
in India. Mathematics began there. You are even today counting 1, 2, 3, etc. to zero, after
Sanskrit figures, and you all know that algebra also originated in India". It is, therefore, very
appropriate that the college, which is named after Swami Vivekananda, is organizing this
conference on Mathematics coinciding with the 150th birth anniversary of Swami
Vivekananda.
2. This conference is most appropriately timed for several reasons. First, the year 2013
has been designated by the International Mathematical Union as The Year of Mathematics of
Planet Earth. Second, we have recently celebrated the 125th birth anniversary of one of
India's greatest mathematics geniuses, Srinivasa Ramanujan (1887-1920). Ramanujan's
true legacy is the demonstration of how a humble, untutored village boy, with sheer passion,
inspiration and inherent talent, rose to become one of the world's greatest mathematicians.
From this perspective, this conference is perfectly positioned to create awareness amongst
the undergraduates about the collaborative nature of mathematics, technology, commerce,
science and social sciences and the vast opportunities that mathematics can offer. In this
background, I plan to talk briefly about the attractions of mathematics, the need for studying
mathematics, its applications, the challenges in mastering the subject, prospects it holds for
those who pursue the discipline and about the risk of misusing mathematics.
Mathematics, the prince charming
3. What is Mathematics?
a. The word itself comes from Greek "Mathema" meaning knowledge, study, learning.
b. There seems to be no consensus among professionals on the issue of defining what
Mathematics is. In fact, you will be surprised that there is even no consensus on
whether it is an art or a science.
c. Definition-wise, there are three sets of definition that I am aware of:
i. Logicist: Benjamin Pierce's "the science that draws necessary conclusions",
or Russell's "all mathematics is symbolic logic".
ii. Intuitionist: From the philosophy of Brouwer "Mathematics is the mental
activity which consists in carrying out constructs one after the other".
iii. Formalist: Identifying mathematics with all its symbols and the rules for
operating on them
4. Mathematics is, perhaps, the oldest of sciences that has existed, developed and
matured in either explicit or in latent form over thousands of years. The concept of numbers
was not only known to prehistoric man but may also be known to animals. For example,
when lions hear a neighboring pride roaring, they calculate how many lions are roaring
compared to the number of lions in their own pride. If there are more lions in their own pride,
or the numbers are equal, or the other pride outnumbers them by up to three to one, they will
always roar back. If the other pride outnumbers them by more than three to one, they stay
quiet. Most animals would know the difference between a bowl containing 4 apples and one
with 8 apples or even the fact that there is something common between 4 apples and
4 oranges.
5. The beauty and charm of mathematics has lured, intrigued and inspired countless
geniuses across the globe to spend sleepless nights in the hope of unraveling its mysteries.
Why have the seekers of knowledge been attracted to mathematics from time immemorial? I
feel the primary charm of mathematics is that it is both interesting and, if you can crack its
intricacies, enjoyable. People like its challenge, its clarity, and the fact that in solving
problems of mathematics you know when you are right. The study of mathematics can satisfy
a wide range of interests and abilities. It helps develop one's imagination and aids in building
a clear and logical thought process.
6. Let me share with you an example to illustrate the charm of mathematics. It is a
well-known story about Ramanujan and his mentor, another famous mathematician, Prof.
G. H. Hardy, who recognized his immense talent and took him to England. At one point of
time, Ramanujan was unwell and lying in an England hospital where Hardy had gone to visit
him. Hardy told him that he came in a taxi, the number plate of which had a most
uninteresting number 1729. Ramanujan was very quick in his reply. He said it was one of the
most interesting numbers that one came across. It is the smallest positive integer, which can
be written as the sum of two cubes in two different ways, viz., (12)3 + (1)3 and (10)3 + (9)3.
7. Prof. C. R. Rao, one of the most outstanding mathematicians in the world, once
said, "All sciences are, in the abstract, mathematics", which aptly captures the immense
contribution of mathematics towards the development of other sciences. Perhaps,
recognizing this centuries ago, Gauss termed Mathematics as the "Queen of all Sciences".
Why study mathematics?
8. Just type the words "why study mathematics" in a Google search and you will get
around 67 million results - these many people, institutes or articles trying to see the benefits
of studying mathematics. Study of Mathematics is extremely important for many reasons.
Maths surrounds us in many ways as we go about our everyday life. Let me now tell you why
you should study mathematics:
a. Mathematics makes life simple by quantification. Numbers, units and dimensions
help in comprehending things better and lead to precision and certainty in
measurement and expression.
b. It helps formulate as well as establish measurement standards in respect of
observable phenomena.
9. Take some simple examples:
a. "Ajay is tall or Ajay is strong" - What do you make of such statements. It will make
sense when we compare with some benchmark or measure and quantify it. Only
then do we know what is tall or what is strong. Then things become precise.
b. Suppose your parents tell you, "You have to study very hard" or "you have to study
very very hard" - do these sentences make much sense? Yes, they certainly make.
But, I am sure, "you have to get 80% marks" is something which will give you
greater focus as the goal is unambiguously stated.
Mathematical concepts like measurement, which were applicable earlier only to physical
sciences, are now being made use of in social as well as biological sciences too.
10. Turning to more formal discussion, one of the important features of a scientific
investigation is its fixation with putting numbers to things, by quantification using
mathematical formulae. Often, scientific work is judged more by the quality of its
mathematics than by its empirical content. Let me quote what James Clerk Maxwell said "All
the mathematical sciences are founded on relations between physical laws and laws of
numbers, so that the aim of exact science is to reduce the problems of nature to the
determination of quantities by operations with numbers".
11. Let me cite some classical examples.
(i) First one relates to modeling of astronomical data, whereby you may recall how
logarithmic tables were found useful for analyzing centuries of astronomical
observations collated by Tyco Brahe and put to use by Johannes Kepler to form the
basic laws of Planetary motion in the early 1600s. Kepler's three laws of planetary
motion are: (i) The path of the planets about the sun is elliptical in shape, with the
center of the sun being located at one focus (The Law of Ellipses). (ii) An imaginary
line drawn from the center of the sun to the center of the planet will sweep out equal
areas in equal intervals of time (The Law of Equal Areas) and (iii) The ratio of the
squares of the periods of any two planets is equal to the ratio of the cubes of their
average distances from the sun. (The Law of Harmonies). It is the third law that
involves difficult exponentiation of squares and cubes, which does not yield easily to
the naked eye when the actual data are observed. Many would know how later
Newton's universal law of gravitation could be found equivalent to Kepler's laws of
planetary motion. But it is also no less magical that Kepler's Third Law of planetary
motion can actually be traced back to Napier's invention of logarithms (1614), which
is supposed to be the main force to derive the postulate about the third law almost
twelve years after the establishment of the first two laws.
(ii) Second, I would like to cite the Mendel's law of segregation of Hereditary Characters
(1866), which was rediscovered at the beginning of the last century, without which,
Darwin's theory of evolution could not be explained statistically. Here, I would also
like to underscore that statistics is a major branch of mathematical science but with
a difference. Math seems to be providing rigorous base for deterministic, logical
framework of establishing truth whereas statistics comes out of measurements and
observations with errors.
(iii) The third one is the now well-known work of the French Mathematician, Louis
Bachelier on the study of finance, which is treated historically as the first of its kind.
His Ph. D thesis on the Theory of Speculation (1900) is credited with being the first
such work on the phenomenon of Brownian Motion that was, much later, put into
use to evaluate stock options.
(iv) In modern times, majority of the financial mathematics driven formulae behind
pricing of different financial assets are testimony to how they can be both used and
abused if not verified rigorously based on real life data.
The above examples relate to close approximation to reality with the help of mathematics,
but based on well-established quantifications and measurement standards.
12. Whether we deal with mechanical, electrical or electronic objects such as the light,
fan, TV, car, bicycle or computers - understanding their functioning calls for use of
mathematics. We all perform tasks ranging from simple arithmetic to complex computations
as we deal with money, deposits, insurance, income tax, and so on. Today's society would
not be in existence without the application of mathematics. In The Republic, the great Greek
philosopher Plato presented a profound argument for why mathematics should be required
for all high school and college students. He argued that mathematics and geometry teach
problem-solving skills and an ability to analyze and think. It is also important to study
mathematics because it gives one a different perspective on things. Learning mathematics
involves a different type of thinking that is not addressed in other subjects.
Applications of mathematics
13. While we all have some familiarity with the everyday uses of the elementary aspects
of mathematics, there are far more advanced and complex phenomena in almost all fields of
science, where mathematics is widely used, but often in an unseen and unadvertised way.
Let me give only a few examples of diverse applications of mathematics:
i. Travel by airplane would not have been possible without the mathematics of airflow
and control systems.
ii. The spaceships' journey to the planets could not have been calculated without the
mathematics of differential equations. The stunning pictures of far away planets sent
by Voyager II could not have had their crispness and quality without mathematics.
iii. The advances in development of supercomputers is backed by the application of
mathematical theory which instructs the computer on what is to be done, thereby
allowing it to optimally utilise its capacity for speed and accuracy.
iv. The next generation of software requires the latest methods in Category Theory, a
theory of mathematical structures which has given new perspectives on the
foundations of mathematical logic.
v. Mathematical methods provide the backbone of Statistical theory and methodology
for the analysis of wide varieties of data in Economics, Banking, Finance, Physical
Sciences, Genetics, Biology, and so on.
vi. Body scanners are the expression of subtle mathematics, which makes it possible to
construct an image of the inside of an object from information on a number of single
X-ray views of it. Thus, mathematics is often involved in matters of life and death.
14. These applications have often been developed from the study of general ideas for
their own sake: numbers, symmetry, area and volume, rate of change, shape, dimension,
randomness and many others. Mathematics makes a special contribution to the study of
these ideas, namely, the methods of (a) precise definitions; (b) careful and rigorous
argument; representation of ideas by many methods, including symbols and formulae,
pictures and graphics; (c) means of calculation; (d) obtaining precise solutions to clearly
stated problems, or clear statements of the limits of knowledge. These features allow
mathematics to provide a solid foundation to many aspects of daily life, and to give a
comprehension of the complexities inherent in apparently simple situations.
15. However, while mathematics is applied to both physical sciences and social /
biological sciences, there are certain differences in the application of mathematics to the two
areas. As physical sciences are more exact in nature, mathematics can be readily applied to
them. However, social/ biological sciences are more uncertain and involve elements of
errors. Hence, statistics and probabilities find greater application in these areas. You, as
students, have to decide on the area which is of greater interest to you and pursue
knowledge and excellence in that area.
Maths in Indian context
16. What is India's contribution to the subject of Mathematics? Let me refer to an old
song from the movie "Purab aur Paschim" (lyrics: Indeevar and others) picturised on the
celebrated actor Manoj Kumar, " ......Bharat ka rahne waala hoon, Bharat ki baat sunaata
hoon...." - how many of you have heard? I am sure all of the speakers here and even the
teachers. The song has some excellent lines depicting India's contribution to the subject of
Mathematics. The lines are as under:
"जब ज़ीरो दया मेरे भारत ने, भारत ने मेरे भारत ने, को तब गनती आई,
दुनया
, |
तार क भाषा को भारत ने पहले सखलाई
दुनया
देता ना दशमलव भारत तो, चांद पे जाना था,
यूँ मुिश्कल
|"
धरती और चांद क का, अंदाज़ा लगाना था
दूर मुिश्कल
Loosely translated, extolling the contribution of India to the field of mathematics and
astronomy, the protagonist conveys that "It was only with the invention of zero by Indian
mathematicians that the World could get its number system. Also, without India's contribution
to the decimal system, it would not have been possible to fathom the distance between Earth
and the Moon or to manage a voyage to the Moon".
17. It is necessary to reminisce about the past heady days of Indian mathematics,
beginning from the Vedic period. The ancient Vedas (synthesized about 5000 years ago)
contained Vedanga Jyotisha comprising three parts, one of which pertains to basic math about
arithmetic, algebra, geometry, trigonometry and equations (Sameekaran). India's primacy in
establishing the foundations of math is widely accepted among the scholars and practitioners
in math. It is apt to note what the famous French mathematician Laplace (1749-1827) had
remarked: "It is India that gave us the ingenious method of expressing all numbers by means of
ten symbols, each symbol receiving a place value position as well as an absolute value. The
idea escaped the genius of Archimedes and Apollonius". While the Indian system of counting
has been the most successful intellectual innovation ever made, the most intelligent invention
is that of zero, the inclusion of which added much needed wholeness and completeness to the
system of natural numbers and for making it the very basic building block of the real number
system. Unfortunately, the rich heritage and achievements of our ancient and pre-medieval
Indian mathematicians were sort of rediscovered and then reintroduced from the West in the
regular curricula on mathematics.
18. It is, indeed, very saddening to note that for the past 2000 years India's contribution
in the field of mathematics has been negligible. We should not rest on our past laurels and
allow complacency to set in. We need to re-establish our position of preeminence in the area
of mathematics and mathematical research by contributing new thoughts and concepts to the
body of knowledge on mathematics. Our students and teachers of mathematics and others
associated with the academic world have to contribute significantly to make it happen.
Mathematics is challenging
19. One should, however, remember that mathematics is also one of the most
challenging disciplines. It calls for special skills and mental ability to visualize the mysteries
of the universe through abstract patterns, symbols, structures and formulae, which is
mathematics. As Galileo said many centuries ago, "The great book of nature can be read
only by those who know the language in which it was written. And that language is
mathematics".
20. I must add a word of caution here. I believe that you must have the necessary
aptitude for studying higher mathematics. In case you do not have the numerical aptitude
and liking for numbers, please do not pursue higher studies in mathematics.
Mathematics as a career option
21. Those who qualify in mathematics are in the fortunate position of having a wide
range of career choices. Their abilities (i) to use logical thought; (ii) to formulate a problem in
a way which allows for computation and decision; (iii) to make deductions from assumptions;
(iv) to use advanced concepts, are all enhanced by a mathematics degree course. It is for
this reason that mathematicians are increasingly in demand. With a mathematics degree,
you should be able to try your hand in banking and finance, statistics, engineering,
computers, teaching, econometrics, biometrics, or accountancy with a unique edge that may
not be available to those graduating in other streams.
22. Thus, one of the benefits of studying mathematics is the variety of career options
that open up. A recent survey has shown that graduates in mathematics and computer
science were at the top of the earning lists, six years after graduation. In one such study in
The Wall Street Journal (2009) on the best and worst jobs in the US, it was observed that the
top three out of the best two hundred jobs listed in order of income and other factors were
careers suited for math majors, namely, mathematics, actuary and statistics. Besides other
favorable conditions like indoor working conditions and places free of toxic fumes or noise,
the study also considers pay, which was determined by measuring each job's median income
and growth potential. Another recent survey shows that the top 15 highest-earning college
degrees have a common element: mathematics.
23. Some of the preferred career options for students of mathematics are:
(a) Mathematics proper as teaching assignment as well as research in the theoretical
aspects; (b) Actuarial science that develops applied tools using mathematics and statistics
and using them in finance and insurance; it includes a number of inter-related disciplines,
including probability and statistics, finance, and economics; (c) Computer science, based on
the theoretical foundations of computation and their implementation and application in
computer systems. Students of maths, with their training in logical and precise thinking, are
highly prized in this field. (d) Operations research, developed as an interdisciplinary branch
of mathematics to arrive at optimal decisions to problems in maximizing or minimizing things
like costs or profits. (e) Biomathematics or mathematical biology, also an interdisciplinary
field of study that helps in modeling natural and biological processes using mathematical
techniques and tools. Results have been applied to areas such as cellular neurobiology,
epidemic modeling, and population genetics. (f) Cryptography is the practice and study of
hiding information. Cryptography is considered to be a branch of both mathematics and
computer science.
24. While applied mathematics is appreciated and understood by many people, pure
maths is considered very elitist. Another career and research prospect that has got
tremendous boost of late is computation based analysis, which many purists do not accept
as mainstream mathematics. Actually, many IT professionals who joined in the big leap of
Silicon Valley revolution in the last couple of decades are now increasingly finding this job of
analytics more rewarding and challenging.
Where math can help in central banking
25. Two important areas of application of Mathematics, more specifically Statistics, are
economic modeling and forecasting, and financial mathematics. Central Banks around the
world are engaged in developing a suite of modern macroeconomic/ econometric models of
the economy as a basis for informed macroeconomic, monetary, financial sector and fiscal
policy decisions. It is now of critical importance, particularly, in light of the current challenges
facing the economy. The importance of modeling macroeconomic variables has become all
the more critical because macroeconomic policy formulation needs to anchor its functional
role in developing a full understanding of the economy as well as influencing domestic policy
formulation in a logical framework.
Let me explain what is modeling and why it is important:
It is trying to study and analyse different phenomena. For this - you have to quantify the
phenomenon so as to be able to relate to them. For example, how much space does a box
contain?
We know volume = length X width X height and as the length, width and height can be
measured, we have a simple model to represent the space inside a box.
26. Financial mathematics is a relatively new discipline, rooted in modern economic
thought, yet steeped in the classical intellectual disciplines of chance and uncertainty.
Tracing its origins to the early 1970s, and maybe more so, to the introduction of the personal
computer, financial engineering's early triumphs include the development of structured
mortgage-backed securities (now the biggest bond market) and a rationale for option pricing
- the consequences of which are totally pervasive in modern investing, the markets, and
finance. Central Banking job has been made much more challenging because it needs to
understand, may be ahead of others in a forward looking framework, the finer aspects used
to construct and deploy the financial transactions and processes. These are the mechanisms
enabling the creation/employment of wealth and for the worldwide distribution of well-being
within the constraints and intent of global financial policy.
27. Modeling exercises in macroeconomics have been rendered more complex and
arduous because of complexities and nonlinearities displayed by behavioral aspects of
economic agents. The traditional economic models depended heavily on linearization of
complex economic behavior expressed in simple mathematical terms.
28. Like much of engineering, financial mathematics constructively uses fundamental
mathematical and scientific principles with professional practices to yield products and
processes. Rather than trying to understand the socio-economic interplay of wealth and
well-being, financial mathematics considers a flow of cash (the cash-flow): its exchange, its
contingency, and its value both in a relative and an absolute sense. These could be from the
standpoint of the investor (central bank, insurance company, mutual fund, for example), a
Wall Street dealer, a global bank, or a hedge fund. The flow of cash could be packaged as a
stock, bond, option, swap, or exchange of currency.
29. Unlike positive sciences like physics and chemistry, one has to take a lot of care
when dealing with behavioural sciences like economics and finance. Lots of measurement
and benchmarking issues come up, particularly, in the modern complex world of business
and commerce that await proper quantifications and standardization with precision and rigor.
Many would have read how floating of risky financial products, not tractable by the
established norms of controllable behavioral norms, jeopardized functioning of markets and
the global economy. With the expanding scope of business and finance, demand for
mathematical acumen and empirical analyses have become ever increasing. However, we
need to guard against utter predominance and capture of the finance profession by the
students of mathematics.
Problems with numbers - life in a central bank is as challenging as is the real life
30. Now that I have shared with you the uses of mathematics in Central Banking, let me
also discuss some challenges/difficulties that the "use of numbers" pose in our day-to-day
office work vis-a-vis the general perception prevalent in the public domain due to improper
interpretation of maths. Let me give three examples:
a. Gold Purchases: You may have come across news reports about the large current
account deficit facing the country and the large import of gold being one of the
important reasons for this. We often hear the argument that people buy gold as a
hedge against inflation or that they are investing in a "safe asset". These people
use, or should one say, misuse mathematics to buttress their argument by relying on
the figure of gold price appreciation in the recent past. However, to me, the data on
gold price appreciation is the most convincing argument for why investing in gold is
neither a hedge against inflation, nor a safe asset. Let me explain. What is the
characteristic of a hedge against inflation? - it should protect your principal by giving
a return slightly above the inflation rate. However, the rate of gold price appreciation
in the recent past has been far in excess of the inflation rate and, hence, cannot be
characterized as a hedge. In contrast, it can be termed as speculation against
inflation. Similarly, the fundamental principal of risk-return trade-off states that
greater returns can be achieved only by assuming greater risks. The significantly
higher returns offered by gold in the past few years only indicates that the risks
implicit in investing in gold have also significantly increased. Even if we calculate
volatility in the gold prices over a longer time horizon, it would be far in excess of
that observed in other financial assets. Hence, the rationale for investing in gold as a
"safe asset" is contrary to conventional wisdom of what constitutes a "safe
investment". Mathematics disapproves that gold is a hedge against inflation or that it
is a safe asset. Unfortunately, this is not fully understood either by investors in gold
or even a significant section of opinion makers and policy makers. We have no
problem if proponents of gold encourage gold purchases by portraying it as a
"speculation against inflation" or a "risky investment" (i.e. right use of mathematics)
rather than by calling it as a "hedge against inflation" or a "safe investment"
(improper use of mathematics).
b. Productivity in Banks: In banks, one of the most commonly used measures of
productivity is Business per Employee. However, any student of mathematics having
some basic understanding of the concept of unit and dimension will say that
Business per Employee may be a good measure of productivity across space but a
very poor measure of productivity over time. If we use this ratio as a measure of
productivity over time in banks/financial institutions for deciding manpower issues,
viz. recruitment of staff, promotions, etc., consequences would not only be
erroneous but can also be dangerous. Even in deciding on the number of General
Managers (GMs) or Executive Directors (EDs) to be provided to banks, policy
makers are depending on the volume of business. Can we not decide on these
issues in a better way by proper use of mathematics, say, based on staff expenses
per 100 rupees of asset/ business or salary paid to GMs/EDs as a percentage of
total assets/business of banks?
c. Financial inclusion and numbers: I am sure you have heard of our initiatives
towards financial inclusion, the business correspondents, the basic banking
accounts, etc. Banks often use "number of accounts" and "number of transactions"
as two indicators for measuring progress in financial inclusion. It is common for
banks to claim progress in financial inclusion stating that the number of accounts
opened has gone up by 100% over a period. This use of mathematics to claim
progress in financial inclusion can be terribly misleading. On delving deeper, one
realizes that while 100 accounts in the previous period have, indeed, increased to
200 accounts, there is no substantive progress in terms of banking penetration and
financial inclusion in real terms, since the total number of villages covered, number
of BCs employed, have also increased manifold during the period. The increase in
number of accounts is, thus, merely a reflection of the expanded geographical
coverage and not of any improvement in banking penetration in existing locations.
Similarly, the number of transactions made per month may have gone up from 50 to
100 but, simultaneously, the total number of accounts may also have gone up from
200 to 1000. Thus, this 100% increase in the number of transactions does not
indicate an increase in efficiency or deepening of financial inclusion. Number of
accounts per 1000 population and number of transactions per account are better
mathematical ratios to judge the progress in financial inclusion.
31. The above three examples that I have given based on my day to day office
experience, are only indicative of the irrational choices that could be made, if mathematics,
as a decision making tool, is not properly used. The students of mathematics must, therefore,
be extremely careful as conclusions based on improper use of numbers can lead to adverse
policy decisions.
Conclusion
32. The key message that I want to convey, particularly to the students, is that not only
the past, but also the future of mathematics and mathematicians is glorious and bright. While
mathematics, in its pure form, is scaling greater heights, the horizons of its applications in
various classical and new fields of science are expanding at a fast pace. Besides the
physical and biological sciences, new applications are found in economics, finance, banking
and many other fields. In view of such diverse applications of mathematics, the whole world
needs mathematicians. Internationally, Indian students and scientists are considered to have
a reputation in mathematics. There is a need to nurture and sustain this natural advantage
which will attract bright students to the field of mathematics and also provide them with
numerous career options. However, in view of the numerous applications of mathematics,
students have to identify their areas of interest and develop domain knowledge in that
particular area. While a career in mathematics research could be pursued by those with a
passion for it, others need to focus on building up specialized expertise in their chosen area
of application of mathematics.
33. While concluding, I would always want you to be like the mathematician in the story
that I am going to tell you:
A mathematician, a physicist, and an engineer were traveling through Scotland
when they saw a black sheep through the window of the train.