I just went through my lesson plan for an upcoming lecture on number-sequences and replaced the name “Fibonacci” by the name “Hemachandra”. By the time you finish reading this essay, you’ll know why I did it, and if you’re a teacher, I hope you’ll do it too. [Note added on November 19: I might now go back again and change “Hemachandra” to “Virahanka”; see the Endnotes.]
To the extent that we can reconstruct the story of the famous sequence
from historical sources, the tale starts with the ancient Indian poet and mathematician Pingala (a contemporary of Euclid’s, give or take a century). For Pingala, these numbers arose from exhaustive consideration of the rhythmic possibilities of Sanskrit poetry. If you want a six-beat poetic phrase built out of short (1-beat) syllables and long (2-beat) syllables, how many possibilities are there? The answer turns out to be 13, so that’s the sixth term of Pingala’s sequence. Likewise, if one is playing the tabla, there are 13 different six-beat drumming patterns one can build from 1-beat and 2-beat components. (The 1-beat and 2-beat components are often rendered vocally as “dhin” and “dha” respectively, so that the two most dissimilar six-beat patterns would be the leisurely “dha, dha, dha” and the rapid-fire “dhin-dhin-dhin-dhin-dhin-dhin”.)
In a similar way, Pingala noted that the number of seven-beat patterns is 21, so that was the seventh terms of his sequence.
Pingala’s description of how one can compute new terms of the sequence is a bit terse (“The two are combined”), but later Indian scholars interpreted the phrase to mean that we compute a new term by combining (that is, adding) the two previous terms. For instance, to find the number of eight-beat patterns, we would add the number of six-beat patterns (13) to the number of seven-beat patterns (21), obtaining 34 as the number of eight-beat patterns.
WHY WE COMBINE
To see why this combination process gives us the right answer, notice that we can divide the eight-beat patterns into two categories: those that begin with “dha” and those that begin with “dhin”.
How many drumming patterns of the first kind are there? Each of them consists of a 2-beat “dha” followed by six more beats which can be built of dhas and dhins in any manner, so the number of eight-beat patterns that start with “dha” is equal to the total number of unconstrained six-beat patterns, which is 13.
And how many eight-beat drumming patterns of the second kind are there? Each of them consists of a 1-beat “dhin” followed by seven more beats which can be built of dhas and dhins in any manner, so the number of eight-beat patterns that start with “dhin” is equal to the total number of unconstrained seven-beat patterns, or 21.
Combining the two cases (which together account for all possible eight-beat patterns) we get 13+21, or 34. I believe that the mathematician and tabla player Manjul Bhargava plays all 34 of these patterns at the end of his informative video “Poetry, Daisies and Cobras”, but I’m not sure. Can any tabla aficionados enlighten me?
The terms of the sequence increase exponentially. For instance, the number of hundred-beat patterns is 573,147,844,013,817,084,101 — that’s far too many patterns for us to ever hope to list, yet we can know precisely how many patterns there are in that stupefyingly large collection of possibilities. I find that a bit magical.
Various Indian mathematicians wrote about the problem of counting rhythmic patterns in the centuries that followed Pingala. Perhaps the most notable was the twelfth-century polymath Hemachandra, who was like Pingala a poet and mathematician and also a linguist, philosopher, and political theorist. Hemachandra described the sequence around the year 1140, and he explained the rule for computing terms more clearly than earlier writers had, but he didn’t attach his own name or anyone else’s to the sequence.
THE FIB IN “FIBONACCI”
Several decades later, an Italian kid named Leonardo was born into a merchant family, got exposed to the mathematics of North Africa (which included much of what had been learned in India and the Arab world in the preceding centuries), traveled far and wide to pick up as much math as he could (see Endnote 1), and wrote a popular book about mathematics featuring a somewhat artificial problem about rabbits. The rabbits weren’t intended to be the main attraction, but the exponentially-growing sequence governing the rabbit population (1,2,3,5,8,13,21,…) struck Europeans as new and different, and a meme was born.
Leonardo was known as “Fibonacci” (“Bonacci” being his father’s name and “figlio” being Italian for “son”), so you might suppose that the term “Fibonacci sequence” came into prominence not long after Leonardo wrote about the sequence in his book. But in fact nobody used the term “Fibonacci sequence” until a century and a half ago, when the French mathematician Edouard Lucas decided to name the sequence after his Italian forerunner. (See Endnote 2.)
This is not the only instance of Europeans naming things after themselves or after other Europeans. It seems to be a thing Europeans like to do. Perhaps the most dramatic instance of this tendency is the way the Italian explorer Amerigo Vespuci succeeded in attaching his name to fifteen million square miles of Earth’s surface. But Leonardo himself gave credit where credit was due, at least as regards the main idea of his book: in championing the system for representing numbers that the West uses today, he rightfully credited India and the Arab world. So perhaps it’s time to imitate Leonardo rather than Lucas (see Endnote 3), especially now that people in the U.S. and elsewhere are seeking a more just reckoning with the past as part of a path toward a more just future.
We could do worse than start by decommissioning the phrase “Fibonacci numbers”.
MATH IS ALREADY GLOBAL; HOW CAN WE SPREAD THE NEWS?
There’s never been a better time for terminological make-overs than the current era of widely-available internet search. If you’re worried that renaming the sequence 1,2,3,5,8,13,21,… would cause massive confusion, consider that anyone can look up “Hemachandra sequence” to find out what it means in a matter of seconds. Likewise, search engines a decade or two from now will incorporate the knowledge that “Fibonacci number” is an old-fashioned term for “Hemachandra number”, so a search for either term will yield matches to both.
Perhaps others more historically versed than I will argue that “Pingala number” or some other phrase from India would be more appropriate than “Hemachandra number”. Let’s have lots of arguments like this, and have them in public! (See Endnote 4.)
Mathematics is a deeply global enterprise, and its true history shows ideas circulating around the planet. As Pingala noticed, it’s when we combine things that exponential growth occurs. When young Manjul Bhargava went back to the original writings of Carl Friedrich Gauss and found new ways to apply ideas that had mostly languished in obscurity, was he doing Indian mathematics or German mathematics? Neither, of course. He was doing mathematics, and he was doing what great mathematicians have done for centuries: taking good ideas wherever they can find them. Simplistic phrases like “imposition” and “appropriation” don’t do justice to this rich process of intercultural dialogue. For instance, what can one say about the Chinese mathematician Xu Guangqi who took some forgotten Chinese mathematics and falsely presented it to the Chinese royal court as being of European origin? (See Endnote 5.) Stories like these resist tidy categorization. And it is precisely these sorts of messy stories that are the most effective antidote to a reductive view of mathematics. If we mathematicians want non-mathematicians to see our subject in the way that we do, then the history of our discipline needs to take a more accurate measure of the contributions that people from all over the world have made to contemporary mathematics.
Why on earth should we present a view of mathematics that falsely stresses the contributions of Europeans when the truth is both more interesting and more reassuring?! If our terminology remains Eurocentric, can we blame people for concluding that our discipline needs to be decolonized? Plus: if we want to attract people from around the world to our discipline, shouldn’t we highlight its global nature, especially since that way of putting a “spin” on things is actually truer than the story we’re implicitly telling now via the names we give things?
Math belongs to everyone, but not everyone is getting access to it. There’s hard work to be done in making things right, especially in the realm of education, but there’s also easy work to be done. What I propose is comparatively easy: let’s credit mathematical ideas to the people and civilizations that gave rise to them. Fibonacci and Pascal and others had a good run of it, with their sequences and their triangles and whatnot, and they did important work that mathematical historians should recognize. But it’s time to bring others out from the shadows of history into the light to receive the recognition they deserve. Let’s get to work.
Thanks to Michael Somos.
#1. Did Leonardo know about Hemachandra? I don’t know.
#2. You can learn more about the history of the Fibonacci numbers from Tina Ghose’s Live Science article “What is the Fibonacci sequence?”. For even more background, read Keith Devlin’s book “Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World”, though I have to ask: if we call Leonardo “forgotten”, what adjective should we use to describe Hemachandra?
#3. A close relative of the sequence 1,2,3,5,8,13,21,…, namely the sequence 1,3,4,7,11,18,29,… (in which the same growth-rule is applied to a different “seed”), is frequently called the “Lucas sequence”, after that same Edouard Lucas. I can’t help wondering: when and how did that happen?
#4. One problem with using nomenclature that gives credit to specific people is that you either create a winner-take-all situation or you create an ungainly string of names separated by hyphens. (Laura Ball’s recent Nautilus article “Mathematicians should Stop Naming Things After Each Other” expresses concern about phrases like “the Grothendieck-Hirzebruch-Riemann-Roch Theorem”; maybe in addition to decolonization, mathematical nomenclature needs dehyphenation!) Also, if some hitherto unknown forerunner suddenly pops up, you suddenly have to change the name to acknowledge the newly discovered “winner”.
Inasmuch as the body of knowledge India developed regarding the sequence 1,2,3,5,8,13,21,… wasn’t the work of a single person but rather a product of India’s mathematical culture over the course of centuries, perhaps the Sanskrit word mātrāmeru would be a more appropriate way to acknowledge India’s mathematical culture without focusing unwarranted attention on individuals. I’m using the term “Hemachandra sequence” in my classes because I gather that’s the commonest name used for the sequence in India today; if I’m wrong, please enlighten me! [Note: After this essay was posted, Manjul Bhargava communicated to me through email his understanding that the name “Hemachandra sequence” is rightfully giving way to the name “Virahanka sequence” in India. I’ll share more when I know more.]
#5. This strange piece of pseudo-appropriation is Roger Hart’s reconstruction of what happened with fangcheng, as described in his book The Chinese Roots of Linear Algebra. If you were a STEM major in the West, you probably learned fangcheng when you studied linear algebra. But you were taught to call it Gaussian elimination.
Virahanka numbers predate Fibonacci by at least 600 years
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Another interesting example of famous ‘incorrectly’ named sequence is the Catalan numbers. Of course, anyone who knows even a little history of the Catalan numbers will know that Euler studied them far before Catalan (the name ‘Catalan’ seems to be due to Riordan). But less known is that they appeared even before Euler in work of Mongolian mathematician/scientist Minggatu: although he had no idea about combinatorial significance of these numbers, they just arose as coefficients in certain trigonometric series. See for instance Igor Pak’s nice history of the Catalan numbers: https://www.math.ucla.edu/~pak/papers/cathist4.pdf.
It’s clear that Fibonacci was not the first with these numbers. But it’s not clear that Hemachandra or Virahanka are better names. Why them, and not the much earlier Pingala?
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I’m basing my decision on what Manjul Bhargava told me in a rather long email. I haven’t gotten his permission to excerpt it yet, but I hope I can post it soon!
Pingala had just touched upon the topic of Matra vrutta (Moraes in poetic meters). It was Virahanka who gave have the number sequences. I have given the link to my article in the earlier comment. Virahanka was earlier then Hemachandra. Hemachandra was a commentator Pingala’s work.