Aardra Jithesh's project on diversity in mathematics
Decolonising mathematics: reclaiming non-Europeans mathematics attributed to European mathematicians
What is decolonisation?
Decolonisation in general can be understood as a term that represents the efforts to resist the cultural and social effects of colonisation and the attempts to right the wrongs of the colonisation and racialisation[1].
Why is this important?
Challenging the exclusion of people of colour in history is important to help dismantle the coloniality of knowledge, which can sometimes even be internalised unknowingly by people of colour themselves.
Many people have a notion that the European way is ‘better’, and people’s views about the supposed superiority of European knowledge typically had racist overtones[2]. Sédillot, a European scholar, claimed that not only was Indian science indebted to Europe, but also that the Indian numbers are an ‘abbreviated form’ of Roman numbers, that Sanskrit is ‘muddled’ Greek, and that India had no reliable chronology. While this scholar did not truly understand the depth of Indian mathematics, he still declared:
“On one side, there is a perfect language, the language of Homer, approved by many centuries, by all branches of human cultural knowledge, by arts brought to high levels of perfection. On the other side [in India], there is Tamil (sic) with innumerable dialects and that Brahmanic filth which survived to our day in the environment of the most crude superstitions. [3]” This notion of European knowledge being superior has continued to the modern day.
When it comes to schooling, many want to be taught in British schools and be taught by British teachers. I have done parts of my schooling in the Middle East in a British school, so I myself know how ingrained it is in the minds of many that education from somewhere more ‘white’ is better[4]. But why does this idea exist? One of the reasons is that coloniality outlives colonisation. By decolonising mathematics in higher education, we can begin to undo the coloniality of knowledge, and help people understand that other societies made incredible advancements, and were intelligent too.
By reclaiming non-Europeans mathematics and giving the credit to them, we challenge the Eurocentric bias that has permeated and dominated mathematics pedagogy, and this can lead to many positive outcomes. It allows the teacher to present an education tailored to the students' experience of their social environment which, in contemporary Britain, includes different ethnic groups with their own mathematical heritage. Furthermore, it provides cultural validation for minority students who are always being reminded, even if indirectly by the absence of any reference to it, they have no mathematical tradition. This further combats ingrained notions of inferiority or internalised colonialism in non-Europeans[2]. Finally, challenging the European standard of mathematics, which is also the standard of education in many places including Britain, can help us gain a rich and thorough understanding of all aspects of mathematics.
“The view that Mathematics is a system of axiomatic/deductive truths inherited from the Greeks and enthroned by Descartes and Kant, has been traditionally associated with a cluster of values that reflect the social context in which it originated. Prime among them are an idealist rejection of any practical, material(ist) basis for Mathematics, from which stems the tendency to view Mathematics as a value-free pursuit, detached from any social and political concerns; and an elitist perspective that sees mathematical work as the exclusive province of a pure, high-minded, nearly priestly caste, removed from mundane preoccupations and operating in a superior intellectual sphere. [5]”
European mathematics typically disregards alternative views of mathematics that have been present in other cultures for centuries, but without following the European standard, we can explore other mathematical views and acknowledge the relation between mathematics and a wide range of disciplines that it conventionally ignores including art, music, architecture, linguistics and history.
Some people may argue that mathematics should not be politicised, and the history of mathematics should stay out of the study of mathematics, but they have and always will be intertwined. Mathematics has been integral to the development of society, has functioned in tandem with the political and social changes of the world, and therefore maths and politics have a natural but clear relationship. From the informal stories we learn about how mathematical concepts came to be, to the names of theorems themselves, mathematics was discovered by humans, and no teaching is truly neutral[6]. While it does not have to be a central component of the curriculum, it is instinctive that a little history will be present while teaching.
While there have been many cases of misplaced credit and we can never truly know who the first person to discover something was, it is important to shift the Eurocentric narrative of mathematics and help bring recognition to non-European peoples' achievements. The intention of this project is not to belittle the incredible discoveries of the European mathematicians, but instead to bring light to the forgotten and under appreciated figures in mathematical history.
Who are some examples of people affected?
Pingala
Pingala was a poet, and his book Chandah Shastra, meaning science of poetic metres, contained 'Pascal's triangle', binary numbers and 'Fibonacci numbers'. His book, also known as Pingala Sutras, was a treatise on prosody, the patterns of rhythm and sound used in poetry. In his book, his binary system consisted of long and short syllables, named Laghu (meaning light) and Guru (meaning heavy) respectively[7].
al-Karaji
al-Karaji wrote 'Pascal's triangle' in a book which is now lost, and then later Khayyam studied it. In Iran it is called Khayyam's Triangle. He was a Persian mathematician and engineer. He proved the binomial theorem using proof by induction, and is the earliest known person to have used proof by induction[8].
Jia Xian
Jia Xian used the triangle as a tool for extracting square and cubic roots. The original book by Jia Xian, called Huangdi Jiuzhang Suanjing Xicao, was lost, however, Jia Xian's work was referenced in detail by Yang Hui. Jia Xian studied the triangle and its link to the binomial coefficients and later Yang Hui popularised it; in China it is called Yang Hui's triangle[9].
Madhava
Madhava or one of his followers discovered the series expansion for sine, cosine and arctangent at the Kerala School of astronomy and mathematics. While his works did not survive, they were referred to by other mathematicians.Using infinite series to approximate trigonometric functions was a significant step towards the development of calculus, and he found the exact formula for pi with this method and his arctangent formula. The ideas and methods used by Madhava for computation of the values of sine and cosine functions are essentially the same as the modern algorithms for the computation of the values of these functions[10].
Where are the women?
Women have faced many forms of oppression throughout history, and many did not have the freedom to study and learn mathematics as their male counterparts did. Women in mathematical history who successfully achieved in making strides and advancements in mathematics often had privileges such as socio-economic status to aid their journey. Hypatia was called a witch and tortured, Sophia Germain had to pretend to be a man to present her work, yet both still had more socio-economic advantages than the other women of their time and were able to study mathematics.
Hypatia was a prominent astronomer and mathematician in ancient Alexandria. She was also the first female mathematician whose life and work are reasonably well recorded. Since Hypatia was the daughter of an upper-class mathematician and philosopher, she received the same education as her male peers. She produced work on conic sections and developed the concepts of circles, ellipses, parabolas, and hyperbolas by dividing cones into planes, and also constructed and taught how to operate astrolabes, a device for monitoring astrological events. She was renowned during her life as a great teacher, and she advised Orestes, the Roman prefect of Alexandria. Orestes had a feud with Cyril, who was the violent Christian bishop of Alexandria. Along with wealthy affluent men in Alexandria, Hypatia was seen as a threat, and Cyril incited rumours Hypatia was a sorceress who bewitched Orestes. Understanding nothing of her philosophy, church leaders called her a witch, and she was tortured and murdered. The Alexandrian school, which she had been the head of, shut down and any philosophers who remained in the city after the destruction fled. Orestes disappeared without a trace, either recalled from his post by the emperor or defecting out of fear he would share the same fate as his friend. All of Hypatia’s writing was lost as part of the church’s conspiracy to repress heretical knowledge[11].
Another example is Sophia Germain, daughter of a wealthy silk merchant, who learnt her mathematical knowledge from books in her father's library. In spite of initial disapproval from her parents, Sophia Germain decided to pursue her passion for maths. The École Polytechnique opened when Germain was 18, which she was prohibited to attend due to her gender. However, the system had created an opportunity to allow anyone to read lecture notes and “submit written observations”. To allow her work to be noticed and receive an unbiased response, Sophia Germain used a fake name of former student Monsieur Antoine-Auguste Le Blanc. Later Sophia Germain talked about "the ridicule attached to a female scientist" to Gauss. Sophia Germain ended up making significant contributions to Fermat’s Last theorem[12].
Why?
It is important to note that the Europeans who got the credit most likely did not steal from the people who discovered it earlier, and they themselves believe they were the first. In cases like Fibonacci with his numbers, he himself in his book wrote he was greatly inspired by Indian mathematics and did not claim the mathematics as his own, yet it was another mathematician who named it ‘Fibonacci numbers’ regardless. There were probably cases of stolen credit, but the main point of this project is to emphasise the incredible contributions from the non-European world that are overlooked.
What can we do?
There have already been efforts made to change the names of many of these mathematical ideas, one example being the series of sine, cosine and arctangent. In literature in recent times, they are now more commonly being referred to as the Madhava-Gregory, Madhava-Leibniz and Madhava-Newton series, with Madhava first to show his precedence. Continuing to give credit to the earlier mathematicians will certainly be a step in the right direction. Additionally, giving credit when teaching in lectures and also in textbooks will not only give non-European mathematicians the recognition they deserve, but also help inspire future mathematicians with representation.
Furthermore, we should challenge the notion that the European standard of mathematics is the universal standard of mathematics. We should stop disregarding alternative cultural perspectives of mathematics and understand and embrace it instead.
References
[1] hooks, bell, 2003. Teaching Community: A pedagogy of hope. London: Routledge.
[2] Kanth, R.K. ed., 2009. The challenge of Eurocentrism: Global perspectives, policy, and prospects. New York, NY, USA: Palgrave Macmillan.
[3] L. A. Sédillot, 1873. “The Great Autumnal Execution.” The Bulletin of The Bibliography and History of Mathematical & Physical Sciences, published by B. Boncompagni, member of Pontifical Academy, Reprinted in Sources of Science, no. 10 (1964), New York & London.
[4] Race, R., Ayling, P., Chetty, D., Hassan, N., McKinney, S., Boath, L., Riaz, N. and Salehjee, S., 2022. Decolonising curriculum in education: continuing proclamations and provocations. London Review of Education, 20(1).
[5] Joseph, G., 1986. A non-Eurocentric approach to school mathematics. Multicultural Teaching, 4(2), pp.13-14.
[6] Gutstein, E. and Peterson, B. eds., 2005. Rethinking mathematics: Teaching social justice by the numbers. Rethinking Schools.
[7] Shah, J., 2013. A history of Pingala’s combinatorics. Gan. ita Bharatı: Bulletin Indian Soc. History Maths, 35(1-2).
[8] Yadegari, M., 1978. The Use of Mathematical Induction by Abū Kāmil Shujā'Ibn Aslam (850-930). Isis, 69(2), pp.259-262.
[9] HONG, S.S., 2011. LIU YI AND HONG JUNG HA’S KAIFANGSHU. Journal for History of Mathematics, 24(1), pp.1-13.
[10] Krishnachandran, V.N., Joy, R.C. and Siji, K.B., On Sangamagrama Madhava’s (c. 1350-c. 1425 CE) algorithms for the computation of sine and cosine functions1.
[11] Alic, M., 1981. Women and technology in ancient Alexandria: Maria and Hypatia. Women's Studies International Quarterly, 4(3), pp.305-312.
[12] O'Connor, J.J., Robertson. E.F., 2020. Marie-Sophie Germain. https://mathshistory.st-andrews.ac.uk/Biographies/Germain/