This is the fourth and final article in the series ‘Mathematicians of Kerala’ wherein the author discusses about possible transmission of Mathematics of Kerala to Europe and concludes by highlighting how such great brains have been neglected by the world for so long.

In addition to my discussion, there is a very recent paper (written by D Almeida, J John and A Zadorozhnyy) of great interest, which goes as far as to suggest Keralese mathematics may have been transmitted to Europe. It is true that Kerala was in continuous contact with China, Arabia, and at the turn of the 16^{th} century, Europe, thus transmission might well have been possible. However the current theory is that Keralese calculus remained localised until its discovery by Charles Whish in the late 19^{th} century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission.

A key development of pre-calculus Europe, that of generalisation on the basis of *induction*, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Keralese ‘routes’.

The need for greater calendar accuracy and inadequacies in sea navigation techniques are thought to have led Europeans to seek knowledge from their colonies throughout the 16^{th} and 17^{th} centuries. The requirements of calendar reform were imperative with the dating of Easter proving extremely problematic, by the 16^{th} century the European ‘Julian’ calendar was becoming so inaccurate that without correction Easter would eventually take place in summer! There were significant financial rewards for ‘anyone’ who could ‘assist’ in the improvement of navigation techniques. It is thought ‘information’ was sought from India in particular due to the influence of 11^{th}century Arabic translations of earlier Indian navigational methods.

Events also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were ‘encouraged’ to acquire mathematical knowledge while there.

It is feasible that these observations are mere coincidence but if indeed it is true that transmission of ideas and results between Europe and Kerala occurred, then the ‘role’ of later Indian mathematics is even more important than previously thought.

### Conclusion

I wish to conclude initially by simply saying that the work of Bharatiya mathematicians has been severely neglected by western historians, although the situation is improving somewhat. What I primarily wished to tackle was to answer two questions, firstly, why have Bharatiya works been neglected, that is, what appears to have been the motivations and aims of scholars who have contributed to the Eurocentric view of mathematical history. This leads to the secondary question, why should this neglect be considered a great injustice.

I have attempted to answer this by providing a detailed investigation (and analysis) of many of the key contributions of the Bharatiya subcontinent, and where possible, demonstrate how they pre-date European works (whether ancient Greek or later renaissance). I have further developed this ‘answer’ by providing significant evidence that a number of Bharatiya works conversely influenced later European works, by way of Arabic transmissions. I have also included a discussion of the Indian decimal place value system which is undoubtedly the single greatest Bharatiya contribution to the development of mathematics, and its wider applications in science, economics (and so on).

Discussing my first ‘question’ is less easy, as within the history of mathematics we find a variety of ‘stances’. If the most extreme Eurocentric model is ‘followed’ then all mathematics is considered European, and even less extreme stances do not give full credit to non-European contributions.

Indeed even in the very latest mathematics histories Bharatiya ‘sections’ are still generally fairly brief. Why this attitude exists seems to be a cultural issue as much as anything. I feel it important not to be controversial or sweeping, but it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Bharatiya, contributions shakes up views that have been held for hundreds of years, and challenges the very foundations of the Eurocentric ideology. Perhaps what I am trying to say is that prior to discoveries made in technically fairly recent times, and in some cases actually recent times (say in the case of Kerala mathematics) it was generally believed that all science had been developed in Europe. It is almost more in the realms of psychology and culture that we argue about the effect the discoveries of non-European science may have had on the ‘psyche’ of European scholars.

However I believe this concept of ‘late discoveries’ is a relatively weak excuse, as there is substantial evidence that many European scholars were aware of some Bharatiya works that had been translated into Latin. All that aside, there was significant resistance to scientific learning in its totality in Europe until at least the 14^{th}/15^{th} c and as a result, even though Spain is in Europe, there was little progression of Arabic mathematics throughout the rest of Europe during the Arab period.

However, *following* this period it seems likely Latin translations of Bharatiya and Arabic works will have had an influence. It is possible that the scholars using them did not know the origin of these works. There has also been occasional evidence of European scholars taking results from Bharatiya or Arabic works and presenting them as their own. Actions of this nature highlight the unscrupulous character of some European scholars.

Along with cultural reasons there are no doubt religious reasons for the neglect of Bharatiya mathematics, indeed it was the power of the Christian church that contributed to the stagnation of learning, described as the dark ages, in Europe.

Above all, and regardless of the arguments, the simple fact is that **many of the key results of mathematics, some of which are at the very ‘core’ of modern day mathematics, are of Bharatiya origin**. The results were almost all independently ‘rediscovered’ by European scholars during and after the ‘renaissance’ and while remarkable, history is something that should be complete and to neglect facts is both ignorant and arrogant. Indeed the neglect of Bharatiya mathematical developments by many European scholars highlights what I can best describe as an idea of European “self importance”.

In many ways the results of the Bharatiya were even more remarkable because they occurred so much earlier, that is, advanced mathematical ideas were developed by peoples considered less culturally and academically advanced than (late medieval) Europeans. Although this comment is controversial it may have been the motivation of several authors for neglect of Bharatiya works, however, if this is the case, then opinions based on those attitudes should be ignored. Bharatiya culture was of the highest standard, and this is reflected in the works that were produced.

**Bharatiya mathematicians made great strides in developing arithmetic** (they can generally be credited with perfecting use of the operators),

**(before Arab scholars),**

*algebra***(independent of the Greeks), and**

*geometry***and**

*infinite series expansions***(attributed to 17**

*calculus*^{th}/18

^{th}century European scholars). Also Bharatiya works, through a variety of translations, have had significant influence throughout the world, from China, throughout the Arab Empire, and ultimately Europe.

To summarise, the main reasons for the neglect of Bharatiya mathematics seem to be *religious, cultural* and *psychological*. Primarily it is because of an *ideological choice*. R Rashed mentions a concept of modernism vs. tradition. Furthermore Bharatiya mathematics is criticised because it lacks rigour and is only interested in practical aims (which we know to be incorrect). Ultimately it is fundamentally important for historians to be neutral, (that includes Bharatiya historians who may go too far the ‘other way’) and this has not always been the case, and indeed seems to still persist in some quarters.

In terms of consequences of the Eurocentric stance, it has undoubtedly resulted in a cultural divide and ‘angered’ non-Europeans scholars. There is an unhealthy air of European superiority, which is potentially quite politically dangerous, and scientifically unproductive. In order to maximise our knowledge of mathematics we must recognise many more nations as being able to provide valuable input, this statement is also relevant to past works. Eurocentrism has led to an historical ‘imbalance’, which basically means scholars are not presenting an accurate version of the history of the subject, which I view as unacceptable. Furthermore, it is vital to point out that European colonisation of Bharat most certainly had an extremely negative effect on the progress of indigenous Bharatiya science

At the very least it must be hoped that the history of Indian mathematics will, in time become as highly regarded, as I believe it should. As D Almeida, J John and A Zadorozhnyy comment:

…Awareness is not widespread.[DA/JJ/AZ1, P 78]

R Rashed meanwhile explains the current problem:

…The same representation is found time and again: classical science, both in modernity and historicity appears in the final count as work of European humanityalone…

He continues:

…It is true that the existence of some scientific activity in other cultures is occasionally acknowledged. Nevertheless, it remains outside history or is only integrated in so far as it contributed to science, which is essentially European.[RR, P 333]

In short, the doctrine of the western essence of classical science does not take objective history into account.

Finally, beyond simply alerting people to the remarkable developments of Bharatiya mathematicians between around 3000 BC and 1600 AD, and challenging the Eurocentric ideology of the history of the subject, it is thought further analysis and research could also have important consequences for future developments of the subject.

It is thought analysis of the difference in the epistemologies of 17^{th} century European and 15^{th} century Keralese calculus could help to provide an answer to the controversial issue of whether mathematics should concern itself with proof or calculation. Furthermore, in terms of the way mathematics is currently ‘taught’ D Almeida, J John and A Zadorozhnyy elucidate:

…The floating point numbers were used by Kerala mathematicians and, using this system of numbers, they were able to investigate and rationalise about the convergence of series. So we(DA/JJ/AZ)believe that a study of Keralese calculus will provide insights into computer-assisted teaching strategies.[DA/JJ/AZ, P 96]

(N.B. computers use a floating-point number system.)

Clearly there is massive scope for further study in the area of the history of Bharatiya and other non-European mathematics, and it is still a topic on which relatively few works have been written, although slowly significantly more attention is being paid to the contributions of non-European countries.

In specific reference to my own project, I would have liked to have been able to go into more depth in my discussion of Bharatiya algebra, and given many more worked examples, as I consider Bharatiya algebra to be both remarkable and severely neglected. Furthermore there is scope for significant and important study of the transmission of Bharatiya mathematics across the world, especially into Europe, via Arabic and later Keralese routes. It is clear that there are many more discoveries to be made and much more that can be written, as C Srinivasiengar observes:

…The last word on the history of ancient civilisation will never be said.[CS, P 1]

As a final note, many question the worth of historical study, beyond personal interest, but I hope I have shown in the course of my work some of the value and importance of historical study. I will conclude with a quote from the scholar G Miller, who commented:

…The history of mathematics is the only one of the sciences to possess a considerable body of perfect and inspiring results which were proved 2000 years ago by the same thought processes as are used today. This history is therefore useful for directing attention to the permanent value of scientific achievements and the great intellectual heritage, which these achievements present, to the world.[AA’D, P 11]

*(The articles in this series appeared on the portal of University of St. Andrews.)*

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