Recently, in India, there was great furore about the introduction of an additional quota for Other Backward Classes for reservations. These protests opened up a whole can of worms about why educational and professional institutions do not have proportional representation of different parts of the population. The pro-reservationist argument that if 20% of the populace of the country belongs to a certain social class, so must 20% of the populace of its educational institutions, rings true for its very simplicity.

Reservation actually reminds one of the closely related notion of Affirmative action. The aim of affirmative action is to positively discriminate towards groups or communities that have been discriminated *against* in the past. The obvious goal of affirmative action, of course, is *justice*, justice to communities and groups that have not had a voice so far. Another goal of affirmative action is *discovering talents that would have been suppressed due to discrimination*. The most interesting claim made by proponents of affirmative action, though, is that affirmative action promotes cultural diversity. The arguments for this are so often quoted that they sound trite: different cultures give rise to different perspectives, different ways of approaching a situation, and hence cultural diversity is important in any area. Hence, Proportional representation of all groups is the desired goal of any public institution, whether academic, professional, social or political.

If the argument sounds specious in this extreme form, we can still ask: to what extent is it true? How does cultural background influence the way one approaches a mathematical problem? Does being Asian, American, or African, endow one with particular ways of thinking that help in the solution of mathematical problems. In other words, how does *generic culture* influence *mathematical culture*?

First, what are the ways in which *background* can influence mathematical ability?

- Genetic factors: Even though it may be
*politically incorrect*to argue that genes could be responsible for mathematical superiority, the contribution of genes can definitely not be ruled out on a scientific basis. - Toddler education: The way a child perceives numbers and shapes is often shaped in his/her initial years. The quality and nature of mathematical education in these years can profoundly influence the child’s conception of numbers. Also, the practical situations in which he/she may use numbers at an early age can also play a profound role in the development of the child’s intuition.
- Primary and secondary education: The nature of the curriculum, the course pressures, and the opportunity to express and develop talent in mathematics are definitely importnat determinants of how the child’s mathematical talents are shaped.

The argument for cultural diversity in mathematical institutions can gain strength if we show that different cultures are strong in these respects in *different* ways and to *different* degrees. That is, we need to argue something like: while a person from culture *A* is naturally better equipped for manipulating algebraic symbols, a person from culture *B* has a better feel for geometric figures. The important point, though, is that the arguments must actually be substantiated by hard facts. Let me look at a very clear and very definite *cultural difference* that is famous in the history of mathematics: Shrinivasa Ramanujan.

Shrinivas Ramanujan spent his childhood in Sarangapani Sannidhi Street, Kumbakonam, Tamil Nadu, born to a particular sub-caste of the Brahmins. Taking a simplified view, the Brahmins are the highest in a four-tier caste system in Hindu society, and the professed purpose of Brahmins is sacred learning. Thus, naturally, being from a Brahmin background meant that Ramanujan has the support and backing of his family (this is all in relative terms to the times) to play around with numbers. It also meant that others were willing to give him a more patient hearing and more of a chance to pursue and study mathematics than they would have given somebdy from a lower caste. In fact, it is inconceivable that a Ramanujan would have emerged from a non-Brahmin family at the time.

Thus, one *cultural* factor that played in Ramanujan’s favour was the fact that he was born a Brahmin.

What makes Ramanujan stand out is not just that he was a great mathematician, but that he had a whole lot of intuition and insight into mathematical truths, and could make astounding statements in mathematics even without formal proof. This has been partially explained by the mystic traditions in Hindu society. In Hindu society, the most rigourous of science developed alongside superstition and mythology, so much so that the two were completely intertwined and difficult to separate. The need for proof or rigour never arose; all results were justified by divine intervention. In this milieu, it was natural that Ramanujan tried to *see the God directly* rather than give conclusive and demonstrative proof.

This contrasts sharply with Europe, where the development of mathematics built on the foundations laid by the Greeks. Even though arguments by the Greeks were often flawed, they *laid down the rules for logic* and *attempted to stick to those rules*, very rarely seeking divine inspiration.

Thus, Ramanujan, as the intuitive mystic we know him to be, may not have existed had he been born in Europe. His talents may have led him to become a mathematician, even a first-rate mathematician, but the lure of the rigour of proof may have slowed down the steady stream of intuition which he poured out.

Is it?

It is really hard to say what Ramanujan could have achieved had he been born into a world where mathematical rigour was valued from Day One. It is possible that within this environment, he would have been able to combine his skills of grasping the results directly with the ability to give conclusive proofs, and gone much, much further. But there remains the possibility that he would have gotten himself restricted to a much more mediocre (though possibly happier) life.

Still, we do have one example of how a person’s culture is *likely* to have influenced his approach to mathematics.

However, the example of Ramanujan hardly gives grounds for the introduction of cultural diversity in institutions. Considering that the Indian *culture* could produce hardly any other mathematician, the fact that Ramanujan was produced by it seems scant justification for introducing a quota for the Indian perspective into mathematics.

In fact, what I feel is that mathematical culture is a very different ballgame from social culture. While social culture may have a few influences on mathematical culture, the fact is that mathematical freedom, creativity, expression and modes of thinking may often starkly contrast the level of freedom, creativity and expression and the modes of thinking existent in general society.

China, Japan, and states in the erstwhile Soviet Union, have produced a whole lot of mathematicians. So have countries like Hungary and Romania. What lies behind their success? A bit of genetics, a sound primary schooling system, and the development of a strong *mathematical culture*.

Hungary has a rich tradition of mathematics largely because it gave rise to mathematics greats like Paul Erdos, Bela Bollobas and so on. These people were not only a source of inspiration to others but also led to the creation of an atmosphere of mathematical contests in the country meant to sieve and train mathematical talent. The contests, the journals, the puzzles, all made people *look forward* to doing mathematics, they helped provide a meaning to mathematics.

A country like Russia, with its tremendous emphasis on a *problem-solving culture*, has naturally been producing great mathematicians. China, with its procedures for selecting and identifying talents at a young age and training them, has consistently been performing well at the Olympiads.

The United States has naturally benefitted and recognized talent and pushed it forward. But as such, the United States has very little of mathematical culture to speak of. It has no zeroes, no number systems, no pi’s and no right triangles to its credit. Yet, it has forged to be a leader in mathematics, with its educational institutions being the much-sought destination. Why? Because its educational institutions, at all levels, have sought to create their own mathematical culture. Each individual has brought in personal initiative and ideas towards fashioning a culture.

Arnold Ross kickstarted mathematics education for young kids in the United States, a precursor to United States’ current Olympiad programmes.

India is culturally poor in mathematics, because we Indians have been unable to create a culture for the kind of mathematics that is practised today. We cling to the importance of Vedic mathematics, spouting its cultural role, forgetting the fact that its primary role is only as a more effective means of computation — a goal that is itself outdated. We talk of the mystic element and the natural intuition of Indians in mathematics, with only Ramanujan to boast of, and he too being a moot point. Our ministers make misguided attempts to revive Indian culture by teaching us to treasure the zero and our numeral system and Vedic mathematics and linking all of them to mysticism and tradition.

And yet, the progress that India as a country has been seeing in creating mathematics and mathematicians is due to the mathematical culture created by individuals. We have not achieved cultural diversity, but at least we are on our way to getting a culture. Home Bhabha started off TIFR, a leading institute for mathematical research, where a *culture for mathematical thought* was developed. Alladi Sitaram, with others, started the Institute of Mathematical Sciences in Chennai, and this further fostered the mathematical culture that has been growing in South India since the time of Ramanujan. Seshadri started Chennai Mathematical Institute to propagate mathematical culture to students right from the undergraduate level.

Cultural heritage? Yes. But not the culture of tradition, of ancient values. It is not the invention of the zero 1200 years ago that moves a culture forward, it is the existence of a system for educating, harnessing and nurturing talent that counts. The past is just a stepping stone to the present.

If we seek cultural diversity within mathematical institutions, then the cultural diversity we need to seek is diversity of *mathematical* culture. The first prerequisite, though, for such diversity is that there should exist a mathematical culture in the first place. And to have this mathematical culture, there first of all needs to be an identification and training of mathematical talent.

What deep insights can an illiterate Indian give into mathematics by virtue of being born into a different culture? How does an African, who has no home support for the pursuit of mathematics and has struggled to study from textbooks publised in America, provide a unique cultural perspective to mathematics. How does a specialist hotelier bring new ideas to the study of differential geometry.

Cultural diversity in a community develops once culture gets developed. When the culture is well-set, trends start getting formulated. For instance, suppose a center for mathematics is established in a city, and young students come to that center to discuss and solve mathematical problems. Then, the *cultural direction* of that center may depend on who decides the collection of library books to be stocked in the center, who conducts classes and gives lectures there, and what policies the center follows. If the center decides to promote mathematical contests with a particular flavour of problems, a culture for those problems is likely to develop. If the center decides to recruit people and conduct lectures in a particular area, a culture for that area is likely to develop.

The mathematical culture itself derives from the mathematical institution rather than from any bare-bones culture. It isn’t Chennai that has a mathematics culture, but institutions in Chennai, be they Chennai Mathematical Institute or the Institute of Mathematical Sciences or the Association of Mathematics Teachers of India. It isn’t Cambridge, Massachussetts that has the mathematics culture but MIT and Harvard. A culture rarely comes imported from the past, the culture is created by individuals and institutions, and only then permeates to the city and the community.

Instead of seeking diversity for our mathematical institutions along the lines of social and economic diversity and diversity of generic culture, we need to seek diversity of mathematical culture. But, in the Indian situation, there hardly exists a mathematical culture in the first place. It is thus necessary to *first create a mathematical culture*, then let it diversify into many cultures, and then automatically reap the benefits of the diversity.

Math game for kids at diff educational sites will dramatically improve comprehension and test scores.

cheers

gudipudi

Comment by gudipudi — January 11, 2007 @ 9:09 am