Utility in Ancient Indian Texts

A cartoon that appeared in today's New Indian Express. This clearly shows the bias of the cartoonist. Anything that is Indian is a burden whereas learning things in a western context is the "correct" way?

I will comment on Ancient Indian Mathematics because that is one area where I have made a comparative study. I will just highlight one topic in mathematics and its utility in teaching a complex topic using the works of Indian mathematicians.

Quadratic indeterminate equations (equations of the form x^2-Dy^2=1) is a topic of advanced number theory currently introduced at MSc part I because the methods used to find integer solution to these equations use the continued fractions method of Lagrange (1766). Bhaskara II (1140 CE) gave a Chakravala (cyclic) method which uses elementary algebra to solve such equations. Hermann Hankel calls the chakravala method "the finest thing achieved in the theory of numbers before Lagrange" (Kaye 1919 Pg 337). Hankel worked with great mathematicians like Mobius, Reimann, Weierstrass, and Kronecker. In fact, the impetus for Bhaskara II was provided by Brahmagupta (628 CE) in his book Brahmasputtasiddhanta where he discusses how to find infinitely many integer solutions from a given integer solution. I am not glorifying the contribution but one needs to understand that a study of Brahmagupta's work can help us introduce these topics at the high school level. How can we expect people to read these fantastic works of eminent mathematicians if the media is going to ridicule the government for introducing anything that is ancient? Whose loss is it? We need to evaluate these things and not just analyze from a political angle only.

A question that arises how can I argue that such topics can be introduced at the high school level? Simple, at Raising a Mathematician Training Program (RAMTP 2017, 2018) we introduced Quadratic Indeterminate Equations in RAMTP 2017 using Brahmagupta's approach and easily grasped by the students. In 2018, we provided students with the reading material and they self-studied the topic and solved problems without being introduced to the topic in a formal way. This helped us introduce Bhaskara's Chakravala method to find an integer solution to these equations. Thus, school students were exposed to a topic which they might have never learned and this was possible mainly because we used Brahmagupta's and Bhaskara's elementary algebra methods.

Conclusion:
1) Such cartoons only create low self-esteem among the countrymen of its nation and devoid them of an ocean of knowledge hidden in the texts written by these great ancient scholars.
2) Learning Pythagoras theorem is not glorifying the ancient Greeks.
3) If the pedagogical goals can be served in a better way by using these ancient texts, we should accept it with open arms.
4) I am only commenting on an aspect that I have researched on and abstaining from commenting on other aspects.

Cricket & Family

It was 22nd April 1998, India V Aus played  the 6th Cricket ODI of the series with Newzealand (NZ) as the 3rd team at Sharjah and if India had to qualify for the finals India had to either win or lose and qualify with a better Net run rate than NZ.

Australia batted first and scored 284. India was struggling at 138/4 as VVS Laxman joined Sachin Tendulkar at the crease. This match had a unique disruption due to sand storm and the revised target under Duckworth and Lewis system was 276 in 46 overs. Sachin played a magnificent innings scoring 143 of 131 balls and had a wonderful partnership with Laxman, who made 23 of 34 balls, of 104 runs. When Sachin got out India had scored 242 and qualified for final although lost the match.

This match is celebrated for Sachin's innings but the role played by Laxman is often understated. The support provided by Laxman in the partnership of 102, although he scored only 23, was crucial in helping India reach the final and ultimately winning the series. If you ask an outsider or look at statistics Laxman's innings may not look important in the outcome but if you ask the team or Sachin you will know how instrumental that innings was.

This is how family life is. Every person is important and each one plays a role in the functioning of the same. Equality only undermines this role and makes one think Sachin's knock was more important than Laxman's. Let us not judge and work in the direction of equity and remember the Asian cultures are family based societies and western cultures are based on Max Weber's sociological theories of individual is supreme and hence individualistic societies. Remember outsiders will always tell you how one person is more equal than other but ONLY you know the role each member of the team plays. Let us not blindly follow the west and understand the great virtues & philosophies that makes our culture unique. Our families are the backbone of our culture, our economy and it is important that we protect this fabric because remember OUR FAMILY BASED SOCIETY IS OUR IDENTITY AND ECONOMIC STRENGTH.

What is the similarity between CEOs, politicians and teachers?

Chief Executive Officers (CEOs) are in charge on the company and are equivalent to the captain of a ship. They provide direction to the company and put efforts to accomplish the vision for which the company is formed. There is a tension between achieving short term outcomes and long term outcomes and sometimes achieving these outcomes can be in apparent conflict. In other words, putting efforts to improve long term outcomes may divert CEO's effort from short term outcomes. However, when the benefits will be reaped by the company the current CEO may not be the CEO of the company and also one may not be able to disentangle the effect of outcome to the effort of this CEO or other factors. This means CEOs have incentives to focus on short term outcomes resulting into decisions with a myopic approach. Needless to say this will be sub-optimal in the long run. In order to overcome this situation, CEOs are incentivized by stock options which motivates them to improve the long term health of the organization.

This kind of behavior is not restricted to executives but we find many professionals having a short term approach to their decision making. Let us take the case of politician. A politician is ideally supposed to take decisions which will be socially optimum in the long run but may be tempted to take decisions which will help the politician to stay in power even if it may be detrimental to the society. Politicians are motivated to "show" that they have taken some action to please the voters. If a politician takes some decision which will have effect in the long run, the same politician may not be in power at that time and hence may not be provided due credit for the actions taken which disincentivizes him/her to take decisions in the interest of the society in the long run.

Now if we look at the case of teachers they are required to help the students improve their marks/grades and also improve their thinking skills, higher order skills and educate them in true sense which will help them in the long run. Focusing on grades/marks (short term goals) is necessary but not sufficient. Not improving higher order skills, thinking skills will be suboptimal in the long run and will not create the necessary human capital. The reason again for the myopic behavior observed is because there is a lot of noise in the relationship between efforts taken by the teacher currently and the outcome that will be observed in the future. However, actions taken can be easily mapped into the improvement of grades/marks and such other performance measures.

What can be a possible solution to improve the situation?
I will focus on education because that to me is the most important area that we need to target sooner than later. There is an apparent constraint in the form of time available for teachers to focus on these two aspects. One thing that can be done is segregate these two responsibilities and have two different persons focusing on short term and long term measures of students' objectives. Since the outcome is difficult to measure in the long run, the educational institutions must be a not-for-profit organizations. Structuring educational institutions as for profit will provide incentives to focus on short term outcomes of marks/grades. Mushrooming of coaching institutes to a large extent is due to the myopic outlook of the society as a whole towards education. This problem is aggravated by the lack of education of parents.

Focusing on outcome rather than efforts is not the first best solution because conversion of efforts into outcome is not straightforward. Incentivizing on efforts although ideal is difficult because it is not easy to observe what the person is doing or how the actions are useful because people who are evaluating performance don't have better information than the persons taking actions. Can we allow professionals to improve long term performance even if it means at the foregoing the benefits in the short term? Our position in the years to come will be determined by the answer to this question.

Ps; I have been an instructor in coaching institutes in the past for Chartered Accountancy (CA), middle school, high school, junior college and degree college students. 

What is special about square root of 2?

Mathematics of ancient India can be very useful to understand the motivation behind the development of Mathematics and can help the students understand how applied Mathematics can be. The interesting questions that can be asked what is so special about √2. Why did Boudhayana specifically provide its value? The length of the diagonal of a square is √2 times the side of the square. Why should I know the diagonal of a square? Squares were used for geometrical constructions and had applications in designing sacrificial altars.

For example: Boudhayana's sulbasutra (800 BCE) provides the value of √2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:¹

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ

samasya problem or puzzle to be solved.

dvikaraṇī. means “that which produces 2” or diagonal of a square

pramāṇaṃ means proof or first term in a rule of three sum

vardhayet means increase

saviśeṣaḥ one that remains

The last word is the most important part of the sutra because it tells us that Boudhayana knew that the result will be an approximation and not the exact value of √2.

The prescription in sulbasutra can be translated as:

Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. The increased length is a small amount in excess²

Thus the above passage from the Sulbasutram gives the approximation:

The symbol ≈ instead of = is because of the word saviśeṣaḥ

In fact this method will in one more step obtain: ³

 ,

Where the only numerical computation needed is 1154 = 2[(34) (17) -1] and, moreover, the method shows that the square of this approximation is less than 2 by exactly

This approximation is accurate to eleven decimal places.

Another thing that I observe is how thin were the compartmentalisation of Mathematics. For example root 2 by Boudhayana uses Geometry. However, value of √2 can be also found using Brahmagupta's varga-prakrti (Brahma-sputa-siddhanata). Brahmagupta (628 CE) uses algebra to find √2 by solving the equation  using “the principle of composition”⁴ or bhavana and finding the ratio  for large values of x & y. 

As can be observed the first positive integer solution for (x, y) is (3, 2) and there are infinitely many solutions to this problem. Although, Brahmagupta couldn’t give the solution to all types of this problem. It sowed the seeds for Acharya Jayadeva and Bhaskara II to design Chakravala process to get a general solution. Another solution to the above equation is (17, 12) which corresponds to Boudhayana’s value of 17/12 if we take first 3 terms  (1+1/3+1/12). Another solution is (99, 70) or (577,408). 

1) Can you figure out how to obtain 577/408 in the Boudhayana’s value of √2? 

2) Can you tell why does x/y gives the approximate value of √2? 

3) Why does the approximation improve for larger values of x & y?

It is needless to say that finding √2 is technically a mundane arithmetic exercise and is not interesting.

Learning this way, we have given an altogether different meaning to √2 and also shown how there can be multiple ways to tackle a same problem in Mathematics. 

Conclusion:

This is just one of the many examples where Ancient Indian Mathematics can be used to motivate higher order thinking among school students and break the barriers of conditional thinking and such inter-linkages will be very useful in contemporary times when knowledge is becoming more and more inter-disciplinary. It is very interesting that the ancient knowledge especially in India was multi-disciplinary. 

References:

1 http://self.gutenberg.org/articles/baudhayana_sulba_sutra#Square_root_of_2

2 This last word is translated by some authors as "The increased length is called savi´e¸a". I follow the translation of "savi´e¸a" given by B. Datta on pp. 196-202 in The Science of the Sulba, University of Calcutta, 1932; see also G. Joseph (The Crest of the Peacock, I.B. Taurus, London, 1991) who translates the word as "a special quantity in excess" and also based on a presentation by Prof Kannan at International Conference on History and Development of Mathematics, Jaipur 2013

3 Square roots in the Sulbasutra – David W Henderson http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html#FOOTNOTE

4 Rationale of the Chakravala process of Jayadeva and Bhaskara II (Historia Mathematica, Page 168)

Turning point in Indian Cricket

I was appearing for my grade X exams in March 2001 and something remarkable happened in Indian cricket history. A lot has been said about the Border-Gavaskar Trophy of 2001 but even today I believe it deserves a mention. Australia had won 15 consecutive test matches before this series and was easily the best cricket team at that time with the best bowling attack of all time. The Australian cricket team played aggressive cricket and always looked to win test matches.

The first test at Mumbai was dominated by Australia and won the game handsomely with Indian batsmen struggling to score runs. The silver lining was the young spinner Harbhajan Singh. India was down in the series and Australia super confident with 16 consecutive wins and having dominated India in the 1st test was looking to close the series at Eden Gardens, Kolkata. Australians batted first in all the matches of that series and scored 400+ in the first innings of Kolkata test. India couldn't bat well against McGrath, Gillespie, Kasprowicz and Warne and conceded a lead of 273 runs. Only VVS Laxman managed to score a half-century but more important was the way he was playing. He was playing positive cricket and stroking the ball nicely. India was asked to follow on by Steve Waugh and with no chance of winning the test, Indians were looking at salvaging some pride and going down fighting. India made a decent start and at the fall of first wicket in came Laxman. Laxman was promoted to number 3 because he was the only batsman who was in great form and was the best batsman for the team. He played positive cricket throughout the innings and scored his 2nd Test century and 2nd against Australia and, along with Dravid, played one of the best test innings in the history. India won that test match and was only the 3rd team to do so in more than 1500 test matches. India went on to win the 3rd test at Chennai with a century by Tendulkar and half centuries by openers- Ramesh and Das and half centuries by Laxman and Dravid. Thus, India won the test series and Harbhajan Singh was adjudged Man of the series for his consistent performance and picking up 32 wickets. He was making a comeback to the Indian team in the absence of Anil Kumble who was injured.

The Kolkata test is definitely a turning point in the Indian cricket. No one gave a chance to India in that test match. Even Indian captain Sourav Ganguly didn't believe that the team could win the test match. That win ended the Australian team's winning streak of 16 tests and gave India the belief that it can win from unfavorable circumstances. This can be seen from the ODI Natwest Trophy final against England in 2002 or Adelaide test in 2003 where once again Laxman and Dravid helped India win a test match. That test was a remarkable victory as winning a test outside subcontinent was always a challenge. That innings of Laxman made Very Very Special Laxman and India under Sourav Ganguly went on to have one of the strongest middle order in the history of cricket.

How Culture Influences Ethics?

Ethics represents a set of moral principles that guides behaviour, and culture is a collective manifestation of a group’s preferences and orientation. Hence, we can see that the two concepts remain closely related and often culture shapes ethical values. Hence, culture is a group phenomenon and ethics is an individual phenomenon. The ethical construct to a large extent depends on our interpretations of truth. Our interpretations of truth are influenced by our own culture.

What is ethical to one group will not be considered so by another group. This is the theory of cultural relativists. According to them, there is no singular truth on which to base ethical or moral behaviour for all time. This approach is in contrast to universalism, which holds the position that moral values are same for everyone. Cultural relativists believe that Universalists thoughts are influenced by their values and believe ethnocentric view makes them less tolerant as they are effectively passing judgements about the other culture.

The choices made by individuals are largely influenced by the circumstances in which he/she lives and is also influenced by community, circle of influence, and to some extent by the macro factors. The modern lifestyle is driven by consumerism. Consumerism believes that happiness is obtained by consuming more and more resources. Since GDP is measured in terms of consumption, there is an incentive for all institutional machineries to promote that as a construct to obtain satisfaction. However, consumerism is subject to inconsistencies because of persistence of older traditions and emergence of new reactions to consumerism.

East and West:

The basic difference between Asian countries and Anglo Saxon countries is Asian countries are family based societies whereas the west is an individualistic society. Individual centric societies tend to be driven by contracts and family based society are driven by social contracts (not necessarily legal). Since family is an economic unit, an individual has more responsibilities apart from accumulating wealth and therefore altruism is developed because of his/her interaction with family members and decision making revolves around maximising welfare and not wealth as individual thinks for the family as a whole and not just the individual. However, it is important for the individuals to increase the circle of influence so as to live life holistically.

Role of religion in Ethics:

Ethics is not only a function of how much wealth does a person possess but also a function of what is the position of wealth in his/her life. Since religions like Hinduism and Buddhism believe in rebirth, their ethical values are derived by individuals requirement of being duty bound and follow righteous path. A person can achieve salvation through right action, knowledge of true nature and spiritual devotion. These religions believe in non-attachment and therefore the ethical quotient among the people who follow these religious path can be expected to be high. The problem that cultures face in today’s times is the intermingling of different cultures creating confusion among the people on the path to take. There is a thin line of difference between following one path and respecting others. Not following other paths is not a sign of disrespect.

Ethics and money:

Ethical principles are put to test when a person is confronted with a situation involving money. Greed is one of the main reasons why a person compromises with his/her outlook towards ethics. Hence, ethical dilemmas are well understood when a person has an opportunity to make gains at the expense of righteousness. Here it becomes very critical to understand the role of spirituality in developing ethical values and hence the importance of culture because if culturally a civilisation has a history of spiritual pursuit/philosophical constructs underlying the people’s behaviour, we can expect the subjects to be more ethical in their daily pursuits of life. It is important to understand that pursuit of spirituality is not against wealth generation but against the methods of wealth generation. When people understand the difference between wealth and illicit wealth, ethical dilemmas will cease to exist.

Ethical situation in India:

Whenever we talk about the Hinduism and the 4 pillars of Sanatana Dharma, we always look at the rich past and how ethical the civilisation and at the same was economically prosperous with less income inequalities. According to me, income equality is a very important outcome of ethical business practices. However, in the current scenario, there appears to be an ethical problem in the country with high levels of corruption and poverty. The greed to become rich at the cost of ethical values has led to this situation. It is important to look at the country’s history to understand the current situation. The invasion of India- first by Moghuls and later by British is a very important event to understand the current situation. Especially during the British rule, we were made to believe that our culture is inferior to the west. This was systematically done by the substitution of indigenous education system with the British model of education. This meant after a generation, people felt superior about the western things, systems as they were only learning about the great things of those nations and when we add the dimension of Industrial revolution we will understand that people started to feel that it is the Hindu way of life which is the reason for this poverty. Hence, the wrong notion of inferiority for the nation’s own culture was created successfully and systematically over the period of British rule. With the degradation of culture, the value for ethics was downplayed and since ethics are learnt primarily through demonstration, the generations to come diluted the meaning of ethics and this resulted in pursuit of business with ethical pollution. Hence, there is a need for the revival of Hindu culture that once existed.

Conclusion: Culture plays a very important role in shaping the economy and ethical values of the nation. The world faces a leadership crisis today and India can provide a direction on how to lead an ethical life since it is one of the oldest living civilisations of the world. Cultural identity is as important as economic identity for the nation. It results in sustainable economic development. Economic prosperity at the degradation of culture can result in chaos and identity crisis which will increase unethical practices.

Ethics can survive in societies where the cultural beliefs habitat a sense of solidarity for other human beings.

Women issues- A general mentality problem

Women in general have to face lot of problems and it is believed that in India the problem is even worse. As I hear about more cases of atrocities against women I start introspecting myself as a member of this society. Things are changing but unfortunately it will take at least a generation I believe for the situation to improve. We as a society think regressive to many problems and the solutions we recommend are many times disheartening.

During my childhood, if I was bullied and I complained to my parents, they had a simple solution- face it or don't go to play. Face it was not a difficult solution for my parents because the problem wasn't a severe one. Don't go to play was an easy solution but not an appropriate one. This is how we think of solving problems. Trying to solve problems by avoiding it rather than solving it. Not finding the appropriate solutions to the problem is actually a problem we are facing. Our big problem as a society is not the problem but the outlook towards the problem or the way we want to tackle the problem. As a teacher when I scold a student for not doing home assignments, I am advised to be mild and sometimes not say anything because kids are not strong enough to accept it. A similar attitude issue is when we look for alliances for marriage we don't normally look for love and compatibility between the partners which should head the list but rather we look for physical age (rather than mental age), caste, language, status, etc as criteria. Again our outlook towards a particular aspect is not very sound, although it works in most cases. Things are changing but will take time. Every society evolves and we have definitely changed for the better in most aspects and we will continue to improve our outlook.

The kind of problems women face is very difficult to imagine and they are really brave enough to face what they go through. The number of rapes, molestation issues at workplace or otherwise faced by women are so dreadful leaves a mental scar which is very difficult to heal. We need to change our outlook towards women and the way many males as well as females look at females need to change. We become judgmental, we start becoming suspicious of women. The problem cannot be solved by asking women to curb the way they behave, or protecting them or asking them not to go out alone. If we are going to continue with this, I am afraid we are looking at simple solutions which tries to side step the problem rather than solving the problem. What we really need is a strong policing, enforcement of laws for such crimes. These are the short term solutions. The long run solution has to come through proper education. We have a long way to go as a society in terms of education and till then we will be improving our literacy rate without increase the group of educated individuals.