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Lately I have been amusing myself by working through Euclid’s Elements. Despite studying mathematics in university, teaching it in high school and *occasionally* using it in my software-soaked day job I never got around to reading Euclid.

Euclid is routinely lionized as the wellspring of axiomatic mathematics. Before *The Elements* mathematicians were *clearly* out of control! They were running around developing useful methods, (counting, fractions, roots), and *— gasp — making unjustified assertions!*

Fortunately, *The Elements* put an end to all that and ushered in the endless age of rigorous axiomatic mathematics. I admire mathematical rigor but my tiny brain can only take so much of it before an all-pervading fog of befuddlement sets in. When I’m all fogged up there are only a few options:

- Reread and rework until the fog clears.
- Press on and review later.
- Give up and abase self.
- Take a break.

I’m a lazy S.O.B. so option (4), take a break, comes up more often than it should. One of my favorite ways to break away from mathematics is to read about it’s *long* history. While tracing the history of* The Elements* I came across the writings of C. K. Raju.

C. K. Raju has written a fascinating book: (the) Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE. Raju’s book is a bit hard to get your hands on. It’s not on Amazon but you can use World Cat to find a copy near you.

Raju’s thesis consist of these major points:

- Significant portions of the
*calculus*developed in India long before Newton and Leibniz and Indian methods, particularly series expansions, came into Europe via 16th century Jesuit missionaries. - European notions of rigorous mathematical proof evolved from the needs of the Catholic Church to convert Muslims with
*impressive*iron-clad logical arguments. The old baffle them with bullshit tactic. Raju claims this*theological*attitude worked it’s way into mathematics and resulted in the*bizarre western view*that deduction is superior to observation, experience and induction. - The ultimate source of eastern secular knowledge, (mostly Arab and Indian), was systematically suppressed and “Hellenized” by the Catholic Church. The church claimed all the “good stuff” in Arab texts originated with the ancient Greeks and had been
*merely preserved by Arab copycats.*It just wouldn’t do to credit hated, (remember the crusades), enemies for their good ideas. - Insisting on rigorous proof when teaching mathematics, especially to children, is sterile and stupid.

All of this reads like a mathematical Dan Brown novel and oddly the Catholic Church is once again the villain. I was enjoying Raju’s account until this passage about Kepler:

*Why, after all, was Tycho so secretive about his papers, not even allowing his trusted assistant Kepler to see them? In any case, on Tycho’s sudden death, Kepler obtained not just Tycho’s observations but also the rest of his papers which contained the underlying theory. Being inclined towards heliocentrism, Kepler transformed Nilakantha’s “Tychonic” orbits to a heliocentric frame (a simple transformation). This made Nilakantha’s variable epicycles come out as ellipses. Being a professional astrologer, Kepler was good at making up stories, and he made up the story about how he had arrived at his results using Tycho’s data.*

In other words Kepler is a fraud and he ripped off one of the major discoveries in astronomy, the elliptical orbits of planets, from Indian astronomers. It’s one thing to spin plausible stories about how parts of calculus may have seeped into Europe from unacknowledged sources it’s another thing to posthumously accuse someone of fraud.

What would it take to make Raju’s case? *How about some hard evidence! *What about Tycho’s secret papers, do any of these documents survive and do they contain references to Nilakantha? Now that would be a smoking gun. Of course we don’t know of any such papers *but that doesn’t mean they didn’t exist.* Proof by conspiracy is a very powerful inference rule — 9/11 troofers and ufologists swear by it! What about the claim that the transformation from Nilakantha’s variable epicycle** Earth centered system **to a **Sun centered** elliptical orbit system is “a simple transformation.” I rather doubt it’s as simple as claimed and even if the transformation was, to use the most abused word in mathematics — *trivial*, it still misses the point. The major shift was to abandon all pretense of **Earth centered systems** no matter how mathematically sophisticated! Before Kepler astronomers and mathematicians, in many cultures, toyed with the idea that planets orbit the sun. After Kepler everyone had to grow up. Planets do orbit the sun deal with it!

And it was precisely how Newton dealt with it that made calculus something worth fighting over. Newton’s unprecedented and monumental proof that elliptical orbits are a mathematical consequence of the inverse square law of gravity is the dividing line between modern and early science. Nothing like it had ever been done before and even today physics and mathematics students are given to chanting we are not worthy when presented with this brilliant argument. Without Newton’s use of the calculus nobody but a few anal mathematicians would give a rat’s ass about who invented calculus.

In a later post I will argue that Raju discounts the importance of independent and coequal mathematical discovery in his account.

“Without Newton’s use of the calculus nobody but a few anal mathematicians would give a rat’s ass about who invented calculus.”

By the same logic no one cares a rats ass about jesus but for the “heathen” (your term not mine) citizens of rome who it duped with divine cross visions and chritinized with leverage of a temporal ruler.

Regarding the initial post, I would like to say that even if an author makes a mistake somewhere in his work, that should not be used to discredit valid parts of his work. There are a large number of extremely interesting ideas in Raju’s work, His history has contemporary relevance to teaching of mathmatics. The mindset that mathematics is about proofs, instead of being about learning how to calculate things is extremely harmful pedagogically. Even if we disagree with one sentence, and find insufficient proof, for it, that should not lead to neglect of his major contributions in other areas.

I completely agree with you. I think C.K. Raju has clearly made his point about Euclid. When the only “biographical” text you have occurs almost a thousand years after the events it allegedly documents you have to question its veracity. I also agree teaching mathematics as one proof after another, especially to beginners, is counterproductive. Where I part with Raju is when he starts going on about limits and religious mathematics. Formal limits came long after infinitesimals and the all the informal techniques, some dating back to Archimedes, that led to calculus. They were introduced by Weierstrass and others to upgrade proofs in analysis; they were never meant to make things easy – only logically sound. Removing limits from mathematics does not make mathematics easy. There are vast branches of mathematics that are limit free and oddly – not easy. As for religious mathematics: I will grant that the Catholic Church went crazy in their zealous suppression of things they didn’t like. The infinitesimal was one of the things that drove them nuts. There is a long paper trail dating back to the early 1500’s showing the church was actively suppressing infinitesimal mathematics. Incidentally this undermines Raju’s speculations that Jesuits helped introduce Indian mathematics to Europe. Why would you spread something you are actively suppressing? In the 16th and 17th centuries you could make a reasonable claim about religious ideas polluting mathematics. This is no longer tenable. Raju makes the point that logic itself is contingent and that there are many logics. Mathematicians have been well aware of this for over a century and have completely incorporated the notion in modern formal theories that, among other things, specify the rules of inference. The things that now count are: is the proof valid given the rules of the system, or is the result useful. This allows plenty of room for both the proof driven and experimental schools.

A great mathematician. Without the inventions the world would be different.

Could some kind Indian historian of Mathematics let me know the names of the Indian astronomers who used the concept of elipse in their workIt is claimed that Aryabhata used the eliptical orbits.I have not come across any citations so far.There are condradictory views about his planetary models.One model uses the mathematical devices like equant,epicycle and deferent and other shows the heliocentric world.Please send me an e-mail if you can.JAY JOLLY.jjjolly2003@yahoo.ca

“While the case for the origin of the calculus in India, and its transmission to Europe is otherwise clear, there remains the important question of epistemology (“Was it really the calculus that Indians discovered?”). For, while European mathematicians accepted the practical value of the Indian infinite series as a technique of calculation, many of them did not, even then, accept the accompanying methods of proof. Hence, like the algorismus which took some five centuries

to be assimilated in Europe, the calculus took some three centuries to be assimilated within the European frame of mathematics.Raju has discussed this question in depth, in relation to formalist mathematical epistemology from Plato to Hilbert, in an article “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa”.

In this paper, Raju proposes a new understanding of mathematics. He argues that formal deductive proof does not incorporate certainty, since the underlying logic is arbitrary, and the theorems that can be derived from a particular set of axioms would change if one were to use Buddhist logic, or, say, Jain logic.

Raju further states, “Indeed, I should point out that my interest in all this is not to establish priority, as Western historians have unceasingly sought to do, but to understand the historical development of mathematics and its epistemology. The development of the infinite series and more precise computations of the circumference of the circle, by Aryabhata’s school, over several hundred

years, is readily understood as a natural consequence of Aryabhata’s work, which first introduced the trigonometric functions and methods of calculating their approximate numerical values. The transmission of the calculus to Europe is also readily understood as a natural consequence of the European need to learn about navigation, the calendar, and the circumference of the earth. The centuries of difficulty in accepting the calculus in Europe is more naturally understood in

analogy with the centuries of difficulty in accepting the algorismus, due, in both cases, to the difficulty in assimilating an imported epistemology. Though such an understanding of the past varies strikingly from the usual “heroic” picture that has been propagated by Western historians, it is far more real, hence more futuristically oriented, for it also helps us to understand e.g. how to tackle

the epistemological challenge posed today in interpreting the validity of the results of large-scale numerical computation, and hence to decide, e.g., how mathematics education must today be conducted.”

(http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm)

Ben,

I am tending towards your view. I agree with some of Raju’s points about giving excessive credit to the ancient Greeks and discounting later discoveries by Asians. However as there is really no hard evidence that Kepler, Newton et cetera were significantly influenced by Hindu mathematics it’s more likely they completely reinvented and greatly extended it. This is not unprecedented in the history of mathematics. Aside from the lunatic “aliens taught us all the good stuff cranks” nobody has seriously argued that the Maya were influenced by eastern or western ideas yet they came up with zero and place value notation all by themselves.

Thanks for stopping by. John

why onus always should be on others to give proof…?

I think it is a matter of thinking straight using elementary common sense.

There was no history of calculus before Newton/ Leibniz period in Europe…whereas there is at least history of 300 (1200 ad to 1500 ad) years of work in India on series expansion/calculus.

so most likely it would have travelled to europe.

Why the hell gregorian Calendar didn’t develop before 1500 ad?

and Indians were making accurate calendar since at least last 2000 years.

so it is time to prove that Newton didn’t borrow from old ideas…

the optics developed during that period was heavily influences by work of Ibn Al haythem…

slowly it is starting to be recognized.

this all need to be looked in the context of crusades, inquisition.

Propaganda is the name of the game.

Thanks for your remarks, to quickly respond to your points:

skeptics always insist that the onus is on the proponents of a proposition to unambiguously demonstrate their point.For example: I will not debate “scientific creationists” until their “theory” makes valid predictions that cannot be accounted for by accepted theories, e.g. evolution. Of course they’ve never produced such arguments so I will ignore them until they do. Life is too short to spend it dueling with dolts.To your other points: “There was no history of Calculus in Europe before Newton/Leibniz…” You are factually wrong and right at the same time.The more correct statement is that Newton and Leibniz were not aware of any “European Calculus but the simple truth is Archimedes developed a form of Integration around 200 BC which predates the Indian Kerala school by 500 or more years. His “method” was very much in the sprit of what C.K. Raju advocates. It was a powerful practical infinite sum method that produced correct results. The volume of the sphere, (a result Archimedes was so proud of it’s alleged it was carved onto his grave), the area under a parabola, the volume of an ellipsoid and so on. These were enormous achievements but Archimedes went further. Unlike C.K. Raju who thinks “proof” is some sort of Catholic conspiracy Archimedes knew his method could not be justified on purely logical grounds. He knew it produced correct results but could not prove it so, in the case of the volume of the sphere; he used the completely sound method of exhaustion to verify the famous volume formula. This is why Archimedes is still counted among the greatest of mathematicians. Unfortunately his “method” was lost to history. The Kerala mathematicians were not aware of it nor were Newton and Leibniz. They knew many of Archimedes results but did not have a clue how he came up with them so they did what many first class mathematicians has always done: they came up with their own methods.I think C.K. Raju completely underestimates the importance of independent rediscovery in mathematics.It happens all the time. Why do modern mathematicians carefully search the literature after working out something they think is new — because it often isn’t? Talented, brilliant individuals working in coherent logical fields frequently come up with similar ideas. The same thing happened all through human history. The simple fact remains there are no clear unambiguous historical documents that demonstrate that Newton or Leibniz were aware of or influenced in anyway by Indian mathematicians. Both men were first class geniuses, so using your thinking straight approach; it makes more logical sense to assume they worked out their results independently. The fact that they came up with completely different notations for the calculus bolsters this point. If both Newton and Leibniz were influenced by the same Indian sources I would expect they would have adapted notations closer to the alleged original source. Of course the Kerala notation is different again. It’s like all these people didn’t post on the same Facebook page!As for the Gregorian calendar, for that we can blame and thank the Catholic Church.Numerous peoples all over the world produced more accurate calendars than the old Julian calendar, (decreed by Julius Caesar and later adapted by the church), perhaps the most accurate are the Mayan and Persian calendars both are still “technically” better than the Gregorian but the Gregorian,viewed as a piece of software is absolutely brilliant.It’s the best calendar ever when you balance accuracy against the complications of inserting intercalary days, (leap days, leap years), to keep things in astronomical synch. The Gregorian scheme of intercalary insertion is much simpler than other accurate calendars and in the modern world of leap seconds further complications are unnecessary. We can curse the church for sticking with the Julian calendar way past its “best before date” but we can also thank them for coming up with what remains a simple and elegant scheme that’s makes the best of our planet’s inconvenient orbit.Finally, let’s stop viewing this through the fractured lenses of propaganda and nationalism.

I’ll say it only once: there is no such thing as Greek, English, European, Indian, Islamic, White, Black, Brown or Asian mathematics. There is only mathematics!Great mathematics transcends it’s culture and I dare say its species!i do know that science does not belong to any Nation, people…etc.

But that was the tacit way in which history of science has been written, and now the course correction is needed.

Honestly tell me that till few years back have you ever heard the name of Ibn Al Haythem?

Tusi?

And it is just not the matter of correction history…

history of science is closely related to the Philosophy of science.

If scientific revolution was not as revolutionary as is stated then probably views of Thomas Kuhn need to be reconsidered.

If we accept that non-western culture came up with non-trivial scientific understanding then,

we need to look at the epistemology behind it, philosophical systemic thought behind it.

This is just not some matter of pride, it is the matter of very contemporary relevance in the matters of practice of science. ( research and education).

The principle contention is contribution by other cultures is not “oh they also ran with us but came distant second, and western science triumphed all the way along”.

But contribution by other cultures was very central to the progress of science.

And original theory so thoroughly ingrained in the mind, that the effect of it is something like propaganda. Onus always left on others to disprove it, however probable the alternate history is.

It takes others multiple generations to come up with some scientific theories, and then out of nowhere the genius from western culture deduce the similar theories, without the priori help.

Atleast some people would start doubting it. And I don’t think there is enough evidence from stopping them from doubting the currently propagated hollywoodified version of history of science.