Between Etymology and Jokes: The Origins of Mathematical Keywords
Curiosity about the origin of a mathematical term spirals into a search that spans centuries. Or it ends with a joke. Either way, the journey could turn more memorable than the equations it was meant to explain.
Numbers and mathematical objects are part of a dictionary between real phenomena and the abstract world, a mapping which has evolved through a global network during millennia.
One way of seeing those connections is through etymology. It provides surprising stories, unexpected links, or tales from the middle of the creative process. Or it could just uncover a joke, with no historical or technical depth.
Algebra Your Bones and The Names of a Polynomial
Many fields of mathematics become clearer as curiosity makes you read more into their history, even if you don't go beyond their names.
The story of algebra for example, both the discipline and the word, is one of the most popular. It sends back to the Arab world of the ninth century AD. That first millennium was rightfully called "The European Dark Ages", since most scientific breakthroughs and mathematical or philosophical writings came from the Indians, the Chinese, and other Eastern peoples.
Muhammad ibn Musa al-Khwarizmi, the main character, was a mathematician at the House of Wisdom, also known as The Great Baghdad Library. It was some time during the 800s when he wrote The Compendious Book on Calculation by Completion and Balancing. In a transliteration of the original title it goes like al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah. The book became known by its short name of Al-Jabr and later through its Latin translation, titled Liber Algebrae. Al-Khwarizmi's foundational role for algebra is well known and the story usually stops here. But what exactly does "al-Jabr" mean? Is it a made-up word? Was it already in the language?
Originally it was a surgical procedure for fixing misaligned bones, due to fractures or joint dislocation. But then it became more generally used for "fixing the alignment of" or "rebalancing" things. Such is the case with al-Khwarizmi's book, which refers to the "realignment" of terms in an equation through cancellation, expansion, factorization, or rewriting. These are now procedures familiar to students solving algebraic equations.
As such, al-Khwarizmi's compendium marks not only the birth of algebra as a discipline of mathematics, but also as a method. It was in his book that the first general and consistent procedures for solving equations appeared. That is, algorithms, another word which is due to him, this time to his name. The algorithm is what all students are familiar to this day for, say, solving a linear equation such as 5x + 1 = 3. Move the free term to the right-hand side by the opposite operations (subtraction), then Divide by the leading coefficient (thus applying once again the opposite operation), and finally, You get the value of the unknown.
Such algebraic expressions are known for millennia through polynomials. Simple examples such as X + 1 or 5X2 + X + 3 are sums of monomials — built by multiplying a number to an indeterminate (X) raised to a power.
But the etymology of "mono-/poly-nomial" speaks about (one or more) names (nomos), quantities or objects. Nouns, if you will. Until the sixteenth century, polynomials were not written symbolically as we do now, but in a prosaic form that referred to practical applications, many geometrical in nature. For example, the first power of X is a length, whereas its second power (X2) is the area of a square — hence the alternative name of "X squared". Raising it to the third power (X3) gives the volume of a cube (again, there's the alternative name of "X cubed").
Poly-nomials have appeared in letters between mathematicians as "stories" like a length and a unit or five squares with a length and three units.
It was the Frenchmen François Viète (aka Vieta) and René Descartes (aka Cartesius), who proposed and consistently used the modern notation, for coefficients, powers, and the indeterminate X. That was not until the sixteenth century, but after that, they revolutionized and simplified how mathematics in general and algebra in particular is written. A focus shift from mathematical objects such as polynomials to their background or container introduces topology. It is a relatively young domain in the history of mathematics, aged just around two centuries. However, a deeper dive into its history shows it emerged much earlier.
Etymologically, "topo-logy" means "the study of place(s)", of space(s), in the most general form. That is, of the background where mathematical objects are created and used. Implicitly, topology also contains the conditions that such background spaces provide for mathematical objects, allowing some to exist, while banishing others.
For example, some high school curricula mention "the topology of the real line", where students learn about intervals, neighborhoods, or accumulation points. Instead of focusing on computations with real numbers, insist on the properties of the underlying set that allow for an infinitude of real numbers in between any other real numbers, to give just an example. It is again etymology which shows why topology has a lot in common with Gottfried W. Leibniz's studies from the seventeenth century, which he called analysis situs: the analysis of place(s).
A couple of centuries later, Henri Poincaré clearly pays tribute to Leibniz by titling his topology treatise Analysis Situs. Poincaré's study contains a lot of modern tools and methods that are used today, but at the same time, title acknowledges the historical depth of this discipline.
Sheaves, Matrices, and Minors
Perhaps polynomials, equations and "spaces" seem familiar, even if you're not a mathematician. But here's a challenge. What kind of image does a 'soft' mathematical object evoke? A soft sheaf to be precise.
Sheaf theory, a modern and highly technical domain of mathematics, developed in close relation to topology. In a very simplified way, think of it as a multitude of properties of the same space, which are layered and connected in some way, like in a sheaf or bunch. I intentionally avoid the word 'bundle' here, as it refers to a distinct mathematical object which I'll explore some other time. In French, where sheaves originate, name (faisceau) shows again they were meant to represent an abstract bouquet of sorts. But the intuition stops here.
No matter where you look, you'll need quite a rich imagination and some humor to see how a sheaf could be soft. Or rather, how softness could be understood mathematically. And there's more: Alexander Grothendieck, one of the most important researchers of the previous century, and his equally genial collaborators such as Pierre Deligne and Jean-Pierre Serre, defined soft, flabby and perverse sheaves, each with its own rigorous definition and complex applications which are difficult to explain outside an advanced program in mathematics.
The names, however, must not be understood using etymology. They're just jokes that Grothendieck and his peers found funny. They may have started in a twisted-metaphorical-vaguely-intuitive notion, but then they evolved to pure fun. Sheaf theory is not the only discipline whose objects are infused with metaphors and fun. Another word that is very popular in multiple areas including many from mathematics is matrix. The Latin origin (...matrix, actually) means pregnant female, but mathematics and science use it for arrays of numbers or other objects, presented in a rectangular arrangement with rows and columns. There's also the related term for factories and production areas, where a matrix is a mold used to shape liquid metal, plastic or some other material cast in it.
Both matrices and pregnant women give rise to new forms — similar but smaller or sometimes identical copies. Historically speaking, that's the actual origin of the term in mathematics. James Joseph Sylvester wrote in 1850s about rectangular objects which are not only useful in themselves, but also as "matrices" (Matrix, his term, which he introduced on the spot) that form other smaller objects.
A rather technical but fitting note is that Sylvester was in fact referring to what we now call determinants. Perhaps surprisingly and against the teaching method of our days, determinants appeared before matrices, as they were useful in computational tasks such as systems of equations. This also explains their names, referring to such systems being over-, under-determined or just determined by the difference between the number of independent equations and unknowns. As such, "the smaller objects" that Sylvester mentioned are parts of determinants which can be computed separately, as smaller determinants themselves. In technical terms, the smaller items are equally appropriately called minors, as emerging from (the determinant of) a larger matrix.
However, even after this historical and etymological excursion I would argue a dictionary is not the most appropriate resource for learning mathematics. But indeed such curiosity go a long way into showing the flow of ideas and discoveries. Some could unfold over centuries or millennia, while others could lead to plain jokes or at least metaphors. You never know what you're going to find and, as in many other cases, the journey is treasured more than the destination.
Axioms, definitions, and theorems may be what we need for exams, but mathematics is much more than that. Sometimes, it really is the occasional metaphor or punchline that teaches us the most.
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