## My All 2020 Mathematics A to Z: Fibonacci

Dina Yagodich suggested today’s A-to-Z topic. I thought a quick little biography piece would be a nice change of pace. I discovered things were more interesting than that.

# Fibonacci.

I realized preparing for this that I have never read a biography of Fibonacci. This is hardly unique to Fibonacci. Mathematicians buy into the legend that mathematics is independent of human creation. So the people who describe it are of lower importance. They learn a handful of romantic tales or good stories. In this way they are much like humans. I know at least a loose sketch of many mathematicians. But Fibonacci is a hard one for biography. Here, I draw heavily on the book Fibonacci, his numbers and his rabbits, by Andriy Drozdyuk and Denys Drozdyuk.

We know, for example, that Fibonacci lived until at least 1240. This because in 1240 Pisa awarded him an annual salary in recognition of his public service. We think he was born around 1170, and died … sometime after 1240. This seems like a dismal historical record. But, for the time, for a person of slight political or military importance? That’s about as good as we could hope for. It is hard to appreciate how much documentation we have of lives now, and how recent a phenomenon that is.

Even a fact like “he was alive in the year 1240” evaporates under study. Italian cities, then as now, based the year on the time since the notional birth of Christ. Pisa, as was common, used the notional conception of Christ, on the 25th of March, as the new year. But we have a problem of standards. Should we count the year as the number of full years since the notional conception of Christ? Or as the number of full and partial years since that important 25th of March?

If the question seems confusing and perhaps angering let me try to clarify. Would you say that the notional birth of Christ that first 25th of December of the Christian Era happened in the year zero or in the year one? (Pretend there was a year zero. You already pretend there was a year one AD.) Pisa of Leonardo’s time would have said the year one. Florence would have said the year zero, if they knew of “zero”. Florence matters because when Florence took over Pisa, they changed Pisa’s dating system. Sometime later Pisa changed back. And back again. Historians writing, aware of the Pisan 1240 on the document, may have corrected it to the Florence-style 1241. Or, aware of the change of the calendar and not aware that their source already accounted for it, redated it 1242. Or tried to re-correct it back and made things worse.

This is not a problem unique to Leonardo. Different parts of Europe, at the time, had different notions for the year count. Some also had different notions for what New Year’s Day would be. There were many challenges to long-distance travel and commerce in the time. Not the least is that the same sun might shine on at least three different years at once.

We call him Fibonacci. Did he? The question defies a quick answer. His given name was Leonardo, and he came from Pisa, so a reliable way to address him would have “Leonardo of Pisa”, albeit in Italian. He was born into the Bonacci family. He did in some manuscripts describe himself as “Leonardo filio Bonacci Pisano”, give or take a few letters. My understanding is you can get a good fun quarrel going among scholars of this era by asking whether “Filio Bonacci” would mean “the son of Bonacci” or “of the family Bonacci”. Either is as good for us. It’s tempting to imagine the “Filio” being shrunk to “Fi” and the two words smashed together. But that doesn’t quite say that Leonardo did that smashing together.

Nor, exactly, when it did happen. We see “Fibonacci” used in mathematical works in the 19th century, followed shortly by attempts to explain what it means. We know of a 1506 manuscript identifying Leonardo as Fibonacci. But there remains a lot of unexplored territory.

If one knows one thing about Fibonacci though, one knows about the rabbits. They give birth to more rabbits and to the Fibonacci Sequence. More on that to come. If one knows two things about Fibonacci, the other is about his introducing Arabic numerals to western mathematics. I’ve written of this before. And the subject is … more ambiguous, again.

Most of what we “know” of Fibonacci’s life is some words he wrote to explain why he was writing his bigger works. If we trust he was not creating a pleasant story for the sake of engaging readers, then we can finally say something. (If one knows three things about Fibonacci, and then five things, and then eight, one is making a joke.)

Fibonacci’s father was, in the 1290s, posted to Bejaia, a port city on the Algerian coast. The father did something for Pisa’s duana there. And what is a duana? … Again, certainty evaporates. We have settled on saying it’s a customs house, and suppose our readers know what goes on in a customs house. The duana had something to do with clearing trade through the port. His father’s post was as a scribe. He was likely responsible for collecting duties and registering accounts and keeping books and all that. We don’t know how long Fibonacci spent there. “Some days”, during which he alleges he learned the digits 1 through 9. And after that, travelling around the Mediterranean, he saw why this system was good, and useful. He wrote books to explain it all and convince Europe that while Roman numerals were great, Arabic numerals were more practical.

It is always dangerous to write about “the first” person to do anything. Except for Yuri Gagarin, Alexei Leonov, and Neil Armstrong, “the first” to do anything dissolves into ambiguity. Gerbert, who would become Pope Sylvester II, described Arabic numerals (other than zero) by the end of the 10th century. He added in how this system along with the abacus made computation easier. Arabic numerals appear in the Codex Conciliorum Albeldensis seu Vigilanus, written in 976 AD in Spain. And it is not as though Fibonacci was the first European to travel to a land with Arabic numerals, or the first perceptive enough to see their value.

Allow that, though. Every invention has precursors, some so close that it takes great thinking to come up with a reason to ignore them. There must be some credit given to the person who gathers an idea into a coherent, well-explained whole. And with Fibonacci, and his famous manuscript of 1202, the Liber Abaci, we have … more frustration.

It’s not that Liber Abaci does not exist, or that it does not have what we credit it for having. We don’t have any copies of the 1202 edition, but we do have a 1228 manuscript, at least, and that starts out with the Arabic numeral system. And why this system is so good, and how to use it. It should convince anyone who reads it.

If anyone read it. We know of about fifteen manuscripts of Liber Abaci, only two of them reasonably complete. This seems sparse even for manuscripts in the days they had to be hand-copied. This until you learn that Baldassarre Boncompagni published the first known printed version in 1857. In print, in Italian, it took up 459 pages of text. Its first English translation, published by Laurence E Sigler in 2002(!) takes up 636 pages (!!). Suddenly it’s amazing that as many as two complete manuscripts survive. (Wikipedia claims three complete versions from the 13th and 14th centuries exist. And says there are about nineteen partial manuscripts with another nine incomplete copies. I do not explain this discrepancy.)

He had other books. The Liber Quadratorum, for example, a book about algebra. Wikipedia seems to say we have it through a single manuscript, copied in the 15th century. Practica Geometriae, translated from Latin in 1442 at least. A couple other now-lost manuscripts. A couple pieces about specific problems.

So perhaps only a handful of people read Fibonacci. Ah, but if they were the right people? He could have been a mathematical Velvet Underground, read by a hundred people, each of whom founded a new mathematics.

We could trace those hundred readers by the first thing anyone knows Fibonacci for. His rabbits, breeding in ways that rabbits do not, and the sequence of whole numbers those provide. Fibonacci did not discover this sequence. You knew that. Nothing in mathematics gets named for its discoverer. Virahanka, an Indian mathematician who lived somewhere between the sixth and eighth centuries, described the sequence exactly. Gopala, writing sometime in the 1130s, expanded on this.

This is not to say Fibonacci copied any of these (and more) Indian mathematicians. The world is large and manuscripts are hard to read. The sequence can be re-invented by anyone bored in the right way. Ah, but think of those who learned of the sequence and used it later on, following Fibonacci’s lead. For example, in 1611 Johannes Kepler wrote a piece that described Fibonacci’s sequence. But that does not name Fibonacci. He mentions other mathematicians, ancient and contemporary. The easiest supposition is he did not know he was writing something already seen. In 1844, Gabriel Lamé used Fibonacci numbers in studying algorithm complexity. He did not name Fibonacci either, though. (Lamé is famous today for making some progress on Fermat’s last theorem. He’s renowned for work in differential equations and on ellipse-like curves. If you have thought what a neat weird shape the equation $x^4 + y^4 = 1$ can describe you have tread in Lamé’s path.)

Things picked up for Fibonacci’s reputation in 1876, thanks to Édouard Lucas. (Lucas is notable for other things. Normal people might find interesting that he proved by hand the number $2^{127} - 1$ was prime. This seems to be the largest prime number ever proven by hand. He also created the Tower of Hanoi problem.) In January of 1876, Lucas wrote about the Fibonacci sequence, describing it as “the series of Lamé”. By May, though in writing about prime numbers, he has read Boncompagni’s publications. He says how this thing “commonly known as the sequence of Lamé was first presented by Fibonacci”.

And Fibonacci caught Lucas’s imagination. Lucas shared, particularly, the phrasing of this sequence as something in the reproduction of rabbits. This captured mathematicians’, and then people’s imaginations. It’s akin to Émile Borel’s room of a million typing monkeys. By the end of the 19th century Leonardo of Pisa had both a name and fame.

We can still ask why. The proximate cause is Édouard Lucas, impressed (I trust) by Boncompagni’s editions of Fibonacci’s work. Why did Baldassarre Boncompagni think it important to publish editions of Fibonacci? Well, he was interested in the history of science. He edited the first Italian journal dedicated to the history of mathematics. He may have understood that Fibonacci was, if not an important mathematician, at least one who had interesting things to write. Boncompagni’s edition of Liber Abaci came out in 1857. By 1859 the state of Tuscany voted to erect a statue.

So I speculate, without confirming that at least some of Fibonacci’s good name in the 19th century was a reflection of Italian unification. The search for great scholars whose intellectual achievements could reflect well on a nation trying to build itself.

And so we have bundles of ironies. Fibonacci did write impressive works of great mathematical insight. And he was recognized at the time for that work. The things he wrote about Arabic numerals were correct. His recommendation to use them was taken, but by people who did not read his advice. After centuries of obscurity he got some notice. And a problem he did not create nor particularly advance brought him a fame that’s lasted a century and a half now, and looks likely to continue.

I am always amazed to learn there are people not interested in history.

And now I can try to get ahead of deadline for next week. This and all my other A-to-Z topics for the year should be at this link. All my essays for this and past A-to-Z sequences are at this link. And I am still taking topics to discuss in the coming weeks. Thank you for reading and please take care.