Hacker News Reader

Comment by breadwinner | Hacker News Reader

In the West the numerals we use are known as "Arabic numerals" because it came from the middle east. But in the middle east the same numerals are known as "Indian numerals" because that's where they got it from.

Similarly, algebra came from the middle east where it was called "al-Jabr" [1], but they in turn got it from India [2].

The invention of calculus is traditionally attributed to Isaac Newton and Gottfried Leibniz. But this book [3] says calculus was brought into being in India in the 14th century by a mathematician named Madhava.

[1] https://en.wikipedia.org/wiki/Al-Jabr

[2] https://www.theguardian.com/world/article/2024/sep/01/hidden...

[3] https://www.nytimes.com/2024/07/24/books/review/the-secret-l...

View on HN

The Arabic-Indic numerals are especially neat because you can see how they evolved as they spread:

https://persianlanguageonline.com/wp-content/uploads/2023/07...

Years ago I worked on a site which was translated into Arabic. Some of the early partners were adamant that we needed to use the Indic numerals everywhere, so I had localized all of our UI, including some JavaScript widgets and worked with a third-party to localize their embed. A few years later, they had a larger conference with a wider swath of the Arabic-fluent world represented and pretty much everyone said we should use what the English-speaking world calls Arabic numerals because most people were more familiar with them than the Indic numeral variants they didn’t use, and computers were making that even more pronounced.

It's difficult to call Madhava's work calculus as we know it. He and his school certainly made huge contributions to analysis of infinite series, but they did not develop that into a unifying framework. It's important to note that no mathematics is done in a vacuum, if Newton/Leibniz "invented" Calculus, it's by building on top of hundreds of years of work done throughout the world.

Fibonacci himself, who popularised their use, explicitly mentioned that they were from India. Somehow this was muddied in later secondary sources.

I encountered this topic a while, back and had a deep look into it, I will be sharing my insights and formed opinion based on the facts that I encountered.

@Dx51Q I appreciate your perspective, but I'd like to clarify a few points regarding Madhava's contributions to calculus. While it's true that Madhava and his school may not have created a unifying framework like Newton and Leibniz did, their work laid crucial groundwork for what we now consider calculus. Madhava is credited with developing infinite series for trigonometric functions such as sine and cosine, which are equivalent to the Taylor series we use today. For example, his series for sin(x) and cos(x) predate those discovered in Europe by over 200 years[1][2].

His followers, like Jyeṣṭhadeva, further elaborated on these concepts in texts like the Yuktibhāṣā, providing proofs and demonstrating their applications[3][5]. Moreover, Madhava's methods for approximating pi were remarkably accurate, achieving values correct to 11 decimal places, showcasing his advanced understanding of numerical analysis[2][4].

This indicates that he was indeed engaging with concepts foundational to calculus, such as limits and convergence. Thus, while Madhava's work may not fit neatly into the modern definition of calculus, it represents a significant and sophisticated mathematical tradition that deserves recognition as a precursor to later developments in the field.

While we are on this topic, we can stop for a second and ponder on why the ancient Indians needed these mathematical formulation. The answer is astronomy, and thus needing a language/framework to understand the cosmos, i.e. mathematics.

Additionally, I wanted to share some interesting insights about the Jesuit transmission of both calculus and the Gregorian calendar from Kerala to Europe. The Jesuit missionaries were not only spreading Christianity through their work, but they were scholars in their own right and could see the value of the advance mathematics they encountered by the Kerala (India) school of mathematics by madhava and the advanced calendar, more accurate than the julian calendar used in Europe at the time.

Jesuit missionaries, especially Matteo Ricci, were really fascinated by the advanced mathematical knowledge coming from the Kerala school. They connected with local scholars, like Brahmins and Kshatriyas, to learn about their mathematical concepts, including those found in texts like the Yuktibhāṣā and Tantrasangraha [5]. This collaboration was part of the Jesuits' efforts to understand local cultures and improve their missionary work. It’s fascinating to think that this exchange not only contributed to the Gregorian calendar reform in 1582 but also helped introduce key calculus concepts into European mathematics.

As to why this is not common knowledge, it’s partly the British colonial policies that muddied the waters and/or suppressed the source of information. But If one looks at it with time, the evidence is there

Citations: [1] https://www.linkedin.com/pulse/madhava-man-who-taught-trigon... [2] https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama [3] https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/ [4] https://en.wikipedia.org/wiki/Madhava_series [5] https://indicmandala.com/the-kerala-school-european-mathemat...

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.

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.)

> 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.

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.

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

Firstly, your claim about Stillwell's "conclusion" is a misrepresentation. Stillwell makes no such conclusion about the lack of transmission. In fact, he explicitly states that the Kerala school knew these mathematical series before 1540. This selective reading and distortion of Stillwell's work is intellectually dishonest and undermines genuine historical inquiry.

The Jesuits sent to India weren't not your typical bible thumpers; they were highly trained mathematicians and astronomers with a specific mission to study and acquire Indian mathematical and astronomical knowledge. The primary motivation for Europeans to import knowledge from India wasn't mere academic curiosity - it was a matter of practical necessity, particularly in navigation. By the mid-16th century, Europeans were grappling with significant errors in their calendar calculations. The true solar year was about 11.25 minutes shorter than the assumed 365.25 days, an error that had compounded over centuries, leading to serious discrepancies in timekeeping and navigation. Matteo Ricci, the Jesuit astronomer and mathematician in a letter from India to Giovanni Battista Maffei (Italian mathematician) he states that he requires the assistance of an “intelligent Brahmin or an honest Moor” to help him understand the local ways of recording and measuring time.

If one wants a smoking gun—a direct admission of knowledge transfer—is either naïve or deliberately obtuse. Do you also believe that tea plants magically teleported from China to India? The British East India Company's industrial espionage in China's tea industry parallels the Jesuits' activities in India perfectly. Both were covert operations aimed at acquiring valuable "know-how" for economic and strategic gain. Do we have a signed confession from Robert Fortune or his kin admitting to tea espionage?

The cumulating circumstantial evidence isn't just substantial—it's overwhelming. We have documented records of Jesuits studying Indian texts, teaching Indian concepts, and corresponding with European mathematicians (see my other comment for examples). The methodological similarities between Kerala mathematics and later European work, like the striking parallels between Wallis and the Yuktibhasa (15th century), where Wallis (in 17th Century) is using the exact expression and reasoning as given in the Yuktibhasa, aren't coincidences—they're smoking guns.

Your dismissal of this substantial body of evidence goes beyond healthy skepticism. It appears to disregard the complex realities of historical knowledge transfer risks coming off as a deliberate attempt to erase non-European contributions to mathematical history.

>The primary motivation for Europeans to import knowledge from India wasn't mere academic curiosity - it was a matter of practical necessity, particularly in navigation.

If the Europeans needed help from Indians "particularly in navigation", then why did the Indians never navigate to Europe before Europeans navigated to India?

Also, in your opinion, did the Europeans continue to copy math from India after they copied calculus or did European civilization suddenly start discovering most of the world's important new math and physics (including refinements of calculus and most of the math and physics that requires calculus) while Indian civilization suddenly slowed way down in the rate at which it discovered important new math?

> Stillwell makes no such conclusion about the lack of transmission

Stillwell: "It is sad that the Indian series became known in the West too late to have any influence or even to become well known until recently." (p. 184, 3rd ed.)

It's quite revealing and ironic that your approach to disagreement and request for concrete evidence is to make false accusations of intellectual dishonesty, dance around the issue making irrelevant analogies to tea, and use emotional rhetoric about "erasing non-European contributions".

I'm fascinated by non-European contributions to maths, science and philosophy because they're substantial and under-recognized historically, but I want actual evidence. Obviously the Jesuits were transmitting some knowledge to Europe, but if you make the stronger claim that Indian mathematicians laid crucial groundwork for calculus, then demonstrate it. That would be intellectually honest and a far better service to the history of Indian mathematics.

What you mention about Wallis and the Yuktibhasa (actually 16th century) is potentially interesting, but you decline to make a concrete case for it and just make vague statements about "striking parallels". Well, let's hear it.

p.s. you're wrong about tea as well: it was cultivated in Assam for centuries before the British were there - history can be complicated;

I found this paper interesting: https://ckraju.net/Joseph/Manchester-paper.pdf

Transmission of the calculus from Kerala to Europe: "...we propose to adopt a legal standard of evidence good enough to hang a person for murder." See document for (1) motivation, (2) opportunity, (3) circumstantial evidence, and (4) documentary evidence.

Thanks, it's interesting and gives some suggestive information about the motivations concerning navigation and calendrical reform.

It has some severe weaknesses when trying to make the case for influence on mathematics, though.

The authors mention Pell's equation x^2+Ny^2=1, and the method given by Bhaskara II to find solutions x,y for arbitrary N (though others note the method was given earlier by Jayadeva).[0] Bhaskara gives the example of N=61. Fermat, in a letter to his friendly rival Frenicle, also gives the example N=61 as a challenge. Now, if 61 were arbitrary in this context, then this would be something like a smoking gun showing almost certain influence. However, 61 is very special here, because the smallest solutions x and y that satisfy the equation are suddenly much larger for 61 than for any smaller N (they're of order 10^10). The authors of the paper don't mention this crucial detail. Fermat also challenges his pal with N=109, which has similarly huge minimal solutions (order 10^15), and which Bhaskara does not give. Fermat writes that he's chosen smallish values of N, "pour ne vous donner pas trop de peine," so he's obviously chosen these values precisely because they're fiendishly difficult, and that's no doubt why Bhaskara also chose 61 as an example, and that's adequate explanation for why the same value shows up in the two places. Fermat had been working on number theory for some decades prior to this point, and his correspondence shows him working through simpler versions of this and related problems, making many original contributions along the way.[0]

The example the authors give about influence on calculus has similar weaknesses. The example they give involves using the sum of the kth powers to integrate x^k. They write that "the formula had no natural epistemological basis in European mathematics," which almost made me spit out my tea because it suggests the authors haven't heard of Archimedes, who used exacly this method with k=2 to give his quadrature of the parabola, and this was well known in Europe at the time (although his works had only really been rediscovered in Europe in the 1500s). The Arab mathematician Al-Haytham had extended the method with k=1,2,3, and 4 in the 10-11th cenury, Cavalieri in 1635 had calculated up to k=9 and conjectured the integral was (1/(k+1))x^(k+1), before Fermat, Pascal, and Roberval gave their proofs of this general relation.[1][2] This last point especially means I don't understand what the authors of the paper mean when they say "the European mathematicians were unable to prove the formula or provide a rigorous rationale for it within their epistemology." ... They did give proofs, and what they were doing was completely intelligible within the historical development leading up to that time.

These are the points I know about, and they don't fill me with confidence about the authors' other judgements.

[0] Andre Weil 1984 Number Theory: An approach through history (ch.1 section 9 mentions Jayadeva and see ch.2 on Fermat, esp. sections 12 and 13)

[1] Edwards 1979 The Historical Development of the Calculus (p.109ff.)

[2] Stillwell 2010 Mathematics and its History (section 9.1)

The Arabic numerals are called Arabic because the Europeans got them from North African Arabs. They are different to the Indian numerals.

> They are different to the Indian numerals.

Not in any mathematically significant way, except for the big endian little endian swap. In Indian system the units is specified first, then the 10s then the hundreds and so on. Arabs took the same system but they wrote right to left. That's why we write the numbers the way we do.

Reply on news.ycombinator.com