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Going to preface this by saying that it's completely uncontroversial that the Kerala school discovered a lot of foundational mathematical concepts independently

Going to preface this by saying that it's completely uncontroversial that the Kerala school discovered a lot of foundational mathematical concepts independently and in many cases earlier than Europeans, but the evidence for any transmission of those ideas to europe is _extremely_ thin, and you can see the organic development of it mostly within Europe in the textual record after the introduction of Arabic numbers and Arabic mathematical texts to Europe (which of course themselves where _hugely_ influenced by Indian mathematics themselves). You can of course make the argument that colonialist historians are motivated to erase the contributions of foreign mathemeticians, but the _mathematicians themselves_ were not shy about crediting the influence of arabic philosophers and mathematicians, so it would have to be explained why they were fine with crediting al-Khwārizmī but drew the line at crediting Madhava. If they did have direct access to the works of the Kerala school, they'd have developed Calculus much more quickly, that's for sure. It's very possible that there was some extremely vague indirect transmission through word of mouth where the source was obscure even to the mathematicians themselves, but I think it's hard to make the argument that "if not for Madhava, calculus would not have been discovered in Europe." There were many scientists and mathematicans all circling around the same problems and several of them solved aspects of it independently at the same time from different directions. And the idea that _Euler_ only discovered power series with help from Indian mathematics is ridiculous. You can see in his own books and his correspondence with others how he gradually worked them out from first principles over time. There's no mystery as to where it came from. If he had just read about them from a book, he would have used them and not spent several years trying to figure it out. The development of math in Europe was _absolutely_ dependent on the introduction of Indian mathematical ideas through Arabic texts, though.

canfakt 2y ago
The claim of "extremely thin evidence" for the transmission of Kerala mathematics to Europe by Jesuits is far from accurate. In fact, there's a wealth of circumstantial evidence supporting this possibility. Jesuits were present in Kerala from 1540-1670, with many, like Matteo Ricci, being highly trained mathematicians tasked with studying Indian sciences. We have clear documentation of their interest in local mathematics, astronomy, and timekeeping, even incorporating subjects like jyotisa into their curricula. Numerous examples show Jesuits actively gathering and transmitting knowledge, from Ricci's inquiries about Indian calendars to Schreck's astronomical observations sent to Kepler. Their close relationships with the Court of Cochin provided access to valuable mathematical manuscripts, and there's evidence of collaboration with Brahmins in translating Sanskrit works. The Jesuits were strongly motivated by practical needs in navigation and calendar reform. Moreover, Marin Mersenne's extensive correspondence network demonstrates that awareness of Indian mathematical knowledge was circulating in Europe. Intriguingly, there are methodological similarities between Kerala mathematics and later European developments, such as parallels between methods used by Wallis and those in the Yuktibhasa. I believe it's crucial to consider the historical context of knowledge transmission between cultures, which often involved clandestine methods. A prime example is the case of Robert Fortune, a Scottish botanist, who in 1848 undertook a covert mission for the British East India Company. Fortune, disguised as a Chinese merchant from a distant province, infiltrated China's heavily guarded tea-growing regions. His objective was to acquire tea plants and seeds, along with the closely guarded secrets of tea production. Fortune's mission was successful; he managed to remove thousands of tea plants and seeds from China, effectively ending the Chinese monopoly on tea production. This act of industrial espionage had far-reaching consequences, leading to the establishment of vast tea plantations in India and Ceylon (now Sri Lanka), and fundamentally altering the global tea trade. While this example pertains to botany rather than mathematics, it illustrates the lengths to which nations went extract knowledge. (Source: Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press.)
- FooBarBizBazz 2y ago
  > A prime example is the case of Robert Fortune, a Scottish botanist, who in 1848 undertook a covert mission for the British East India Company. For anyone else reading, I'll mention that I found this book extremely engaging: Rose, Sarah. For All the Tea in China: Espionage, Empire, and the Secret Formula for the World's Favourite Drink.
  - srean 2y ago
    Any one who knows his/her tea, opium, coffee and chocolate knows a lot about the world. I don't know enough about cocaine to comment on it. The battle for and between tea and opium is truly bizarre. Today we have war against drugs, the British were fighting for the opposite -- to make China hooked on opium (to counterbalance the import cost of tea) and then blame China -- you opium addled lesser civilization.
- griffzhowl 2y ago
  The thing is, circumstantial evidence that transmission could have happened, though suggestive, isn't positive evidence that it did happen. The absence of such direct evidence such as translations or quotations from relevant Indian works in Europe is even emphasized in Joseph's book that you give as ref. The claim that Madhava "laid crucial groundwork" for the development of calculus in Europe needs to be backed up with some argument saying exactly what was transmitted and when. The problem with simply observing that his results on infinite series for arctan, sin and cos reappeared in Europe much later is that this doesn't rule out independent discovery in Europe. And the case for independent discovery is strong, because we have the documentary evidence of the logic of discovery: Newton and Gregory both obtained these series from the general binomial series, seemingly independently of each other, and to my knowledge Madhava didn't have this result.[0] They're both clearly basing their works on documented predecessors (who happen to be European) such as Wallis, Fermat, Descartes, Viete, and others, so the trajectory of how the results were obtained seems to be accounted for. So what was the crucial groundwork, and when was it transmitted? There might have been something (e.g. your suggestion about parallels between Wallis and the Yuktibhasa), but I haven't seen a strong argument that gives these details. I would be fascinated if there were though. [0] Stillwell 2010 Mathematics and its History (3rd ed.) - Stillwell is sympathetic to the Kerala school, is very clear in assigning Madhava priority on the arctan, sin and cos series, but concludes, along with most historical scholarship, that it seems there was no transmission of these results to Europe

canfakt 2y ago

The claim of "extremely thin evidence" for the transmission of Kerala mathematics to Europe by Jesuits is far from accurate. In fact, there's a wealth of circumstantial evidence supporting this possibility. Jesuits were present in Kerala from 1540-1670, with many, like Matteo Ricci, being highly trained mathematicians tasked with studying Indian sciences. We have clear documentation of their interest in local mathematics, astronomy, and timekeeping, even incorporating subjects like jyotisa into their curricula. Numerous examples show Jesuits actively gathering and transmitting knowledge, from Ricci's inquiries about Indian calendars to Schreck's astronomical observations sent to Kepler. Their close relationships with the Court of Cochin provided access to valuable mathematical manuscripts, and there's evidence of collaboration with Brahmins in translating Sanskrit works. The Jesuits were strongly motivated by practical needs in navigation and calendar reform. Moreover, Marin Mersenne's extensive correspondence network demonstrates that awareness of Indian mathematical knowledge was circulating in Europe. Intriguingly, there are methodological similarities between Kerala mathematics and later European developments, such as parallels between methods used by Wallis and those in the Yuktibhasa. I believe it's crucial to consider the historical context of knowledge transmission between cultures, which often involved clandestine methods. A prime example is the case of Robert Fortune, a Scottish botanist, who in 1848 undertook a covert mission for the British East India Company. Fortune, disguised as a Chinese merchant from a distant province, infiltrated China's heavily guarded tea-growing regions. His objective was to acquire tea plants and seeds, along with the closely guarded secrets of tea production. Fortune's mission was successful; he managed to remove thousands of tea plants and seeds from China, effectively ending the Chinese monopoly on tea production. This act of industrial espionage had far-reaching consequences, leading to the establishment of vast tea plantations in India and Ceylon (now Sri Lanka), and fundamentally altering the global tea trade. While this example pertains to botany rather than mathematics, it illustrates the lengths to which nations went extract knowledge. (Source: Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press.)

FooBarBizBazz 2y ago
> A prime example is the case of Robert Fortune, a Scottish botanist, who in 1848 undertook a covert mission for the British East India Company. For anyone else reading, I'll mention that I found this book extremely engaging: Rose, Sarah. For All the Tea in China: Espionage, Empire, and the Secret Formula for the World's Favourite Drink.
- srean 2y ago
  Any one who knows his/her tea, opium, coffee and chocolate knows a lot about the world. I don't know enough about cocaine to comment on it. The battle for and between tea and opium is truly bizarre. Today we have war against drugs, the British were fighting for the opposite -- to make China hooked on opium (to counterbalance the import cost of tea) and then blame China -- you opium addled lesser civilization.
griffzhowl 2y ago
The thing is, circumstantial evidence that transmission could have happened, though suggestive, isn't positive evidence that it did happen. The absence of such direct evidence such as translations or quotations from relevant Indian works in Europe is even emphasized in Joseph's book that you give as ref. The claim that Madhava "laid crucial groundwork" for the development of calculus in Europe needs to be backed up with some argument saying exactly what was transmitted and when. The problem with simply observing that his results on infinite series for arctan, sin and cos reappeared in Europe much later is that this doesn't rule out independent discovery in Europe. And the case for independent discovery is strong, because we have the documentary evidence of the logic of discovery: Newton and Gregory both obtained these series from the general binomial series, seemingly independently of each other, and to my knowledge Madhava didn't have this result.[0] They're both clearly basing their works on documented predecessors (who happen to be European) such as Wallis, Fermat, Descartes, Viete, and others, so the trajectory of how the results were obtained seems to be accounted for. So what was the crucial groundwork, and when was it transmitted? There might have been something (e.g. your suggestion about parallels between Wallis and the Yuktibhasa), but I haven't seen a strong argument that gives these details. I would be fascinated if there were though. [0] Stillwell 2010 Mathematics and its History (3rd ed.) - Stillwell is sympathetic to the Kerala school, is very clear in assigning Madhava priority on the arctan, sin and cos series, but concludes, along with most historical scholarship, that it seems there was no transmission of these results to Europe