Memorizing words in a foreign language without understanding them feels like a special kind of fluency, one built upon forms without meaning. Mathematics, too, is a language that suffers from a superficial approach focused on rote learning.

Numbers and mathematical objects are part of a dictionary between real phenomena and the abstract world, which has evolved through a huge network of collaborations during millennia.

Etymology provides surprising stories, unexpected connections, or tales from the middle of the creative process. Or you could just uncover jokes, with no historical or technical depth.

Here are two kinds of mathematical terms: some for which meaning is easily found and only gets richer the more you keep digging, and others which emerged from metaphors or simply as a joke.

Algebra Your Bones and The Names of a Polynomial

Many branches of mathematics become clearer as curiosity makes you read more of their history — even when you don’t go beyond their very names.

Algebra, for example, sends you back to the Arab world of the ninth century. The first millennium AD was rightfully called “The European Dark Ages”, since most scientific breakthroughs, as well as mathematical or philosophical writings have come from the Indians, the Chinese, and other Eastern peoples.

Muhammad ibn Musa al-Khwarizmi was a mathematician who worked at the House of Wisdom or The Great Baghdad Library sometime during the 800s, when he wrote The Compendious Book on Calculation by Completion and Balancing, or, in a transliteration of the original title: al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah. The book was known by its short name Al-Jabr and later through its Latin translation, titled Liber Algebrae.

Al-Khwarizmi’s foundational role for algebra is well known, but what exactly does al-Jabr mean? Originally, it was a surgical procedure for fixing misaligned bones, due to fractures or joint dislocation. But then it became more widely 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 or rewriting.

Al-Khwarizmi’s compendium marks not only the birth of algebra as a word, but also as a method. It was in his book that the first general and consistent procedures appeared for solving equations — algorithms, another word which is due to al-Khwarizmi, this time his name.

It is the very procedure used to this day for, say, a linear equation such as 5x + 1 = 3. Move the free term to the right-hand side by changing its sign then divide by the coefficient and you get the value of the unknown.

Algebraic expressions like the one in the equation above are studied with objects known for millennia: polynomials. Simple examples such as X + 1 or 5X² + X + 3 are obtained as sums of monomials — terms made 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), meaning quantities or objects. This is because until the sixteenth century, polynomials were not written symbolically as above, but in a prosaic form that referred to practical applications, many geometrical in nature.

For example, an indeterminate raised to the first power (X) is a length, whereas its second power (X²) is the area of a square — hence the alternative name of “X squared” —, and raising it to the third power (X³) gives the volume of a cube (again, there’s the alternative name of “X cubed”).

Polynomials such as the ones I gave as examples might 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, 1540-1603) and René Descartes (aka Cartesius, 1596-1650) who proposed and consistently used the modern notation, for the coefficients, the powers, and the indeterminate X.

The use of this character in equations is a story for another time, but for now note that in 1557 the Welsh mathematician Robert Recorde was using a totally different notation.

A focus shift from mathematical objects such as polynomials to their background or container reaches topology. It is a relatively young domain in the history of mathematics by some authors, aged just around two centuries. However, a deeper dive into its history shows it emerging 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 the emergence of some objects, while prohibiting others.

For example, the high school curriculum sometimes mentions “the topology of the real line”, where students learn about intervals, neighborhoods, or accumulation points. Instead of focusing on computations with real numbers, focus on the properties of the underlying set that allow for an infinite set of real numbers in between any two real numbers, to give just an example.

Or if you’re interested in the curved shape of the Universe, topology insists on understanding the fundamentals of curvature as a property or certain cuts or paths that are allowed in this space, instead of studying the planets or the light therein.

Such examples show 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. His study contains a lot of the modern tools and methods that are used today, but at the same time, he uses the title to acknowledge the historical depth that this discipline has.

Sheaves, Matrices, and Minors

What kind of image does a ‘soft’ mathematical object evoke? To be precise — a soft sheaf. Sheaf theory, a modern and highly technical domain of mathematics, emerged in close relation to topology. In a very simplified way, think of it as a characterization of multiple properties of the same space, which are layered and connected in some way, like in a sheaf or bunch. I avoid the word ‘bundle’ here, as it refers to a distinct mathematical object which I’ll explore in a future essay. In French, where it originates, it has a similar name, faisceau, showing that it really was meant to represent an abstract bouquet of sorts. But the intuition stops here.

No matter where you look though, you 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) 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 that is used to shape liquid metal, plastic or some other material that is cast in it.

Both matrices and pregnant women give rise to new forms — some that are similar but smaller or sometimes identical copies. Historically speaking, that’s the actual origin of the term in mathematics. James Joseph Sylvester wrote in the 1850s about rectangular objects which are not only useful in themselves, but also as “matrices” (Matrix, his term) which 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 equations and the number of solutions.

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.

Surely a dictionary is not the most appropriate resource for learning mathematics, but curiosity for history and etymology could 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.