Mathematics has a complex, highly-specialised language. Contrary to popular belief, this does not purely consist of a world of formulas, but is instead supplemented by a rich and creative imagery in daily research in order to make vague intuition more comprehensible.

A central requirement of mathematical language is that it leaves absolutely no room for subjective interpretations, since the objects that it references are completely abstract. For a rough analogy, just imagine which challenges could arise in language if we were to communicate with someone who is missing one of their senses. Or how many twists and turns we would have to take to describe effects that are simply impossible to perceive through our senses.

A further challenge here is that mathematical objects, in many cases, behave in a way that goes against our intuition. This problem was already discussed extensively in the early 20th century.1 For a concrete, extreme example, just search for the Banach-Tarski paradox online, or, for something a little more accessible, Hilbert’s Hotel.

Mathematics overcomes the challenges of its language through the axiomatization of its foundations and an intrinsic terminology that stems from it, which is both precise and extensive. Let’s look at some examples — take matrices: They have a rank, which in linear formation has a kernel and an image, eigenvalues that we define and a determinant that we can calculate. They can be positive semidefinite, totally unimodular or nilpotent (and much more). Many of these terms have been borrowed from the real world. In algebra we study for example groups, fields and rings2 — and in rings preferably ideals. We like to name objects after the person who discovered them. Think about Hilbert, Banach or Hausdorff spaces. A good ring is Noetherian. And neither Kummer surfaces nor Zorn’s lemma have anything to do with the emotions of sadness or anger — as a literal translation of the names would suggest. From this terminology comes a language. For example, we define a polytope as a »convex set of finite points contained in the n-dimensional real space«. And even terms that initially appear imprecise usually have a specific meaning – for example that rational numbers are close to real numbers, or that in measure theory a property holds almost everywhere.

Timo de Wolff

Language and images in mathematics

JAM #28 — Image Language

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However, there is also what could be described as »mathematical jargon«: For example, a well-known theorem that is an unpublished result can be referred to as »folklore«. An important or difficult statement is »deep« and a proof which is nicely or creatively worded is »elegant«.

Why then all the intimidating formulas might you ask. The simple answer is: To keep the language precise and verifiable. The alternative can be seen in the prosaic and often hardly legible mathematical texts from prior to the (mid) 19th century.

Formulas help keep facts specific and free from the need for subjective interpretations. And so one of the seven most renowned Millennium problems3 can be narrowed down to just five characters: »P=NP?«.

All of this is completely different in the context of active mathematical research. That is like a path through thick clouds in which we suspect distorted patterns and try to make sense of them. Often that literally means, to stand in front of a board with other people, draw sketches and describe (still) vague intuitive patterns. Precise language is bound to be lacking in this case given that we don’t yet know what we are specifically talking about and first have to create language and make structure more concrete. The actual proof can only take place when structure is recognised and language is sufficiently extensive.

Once a mathematical theory is established, images used to explain it often persist. But how does one imagine these images? Well, let’s finish this article with a quick experiment: Take a rectangle, with random edge lengths and the angles a, b, c, d from the upper left to bottom right. Then we stick the upper edge to the lower one (a to c and b to d) and then the left edge to the right one. We can imagine or test that the object we get as a result will be a 2-torus (an object with a hole in the middle in the shape of a doughnut). But what happens when we, before the first step, turn the upper edge by 180 degrees (so a to d and b to c) and then stick the remaining edges together (without turning them)?
[Solution see below]

1See for example Hans Hahn: »Die Krise der Anschauung« (The Crisis of Perception), in: Empirismus, Logik, Mathematik (Empiricism, Logic, Mathematics), Suhrkamp, Frankfurt am Main (Germany) 1988.

2Not in the sense of the ring that we wear on our fingers, but in the sense of the German word for a »band of people«.

3For more on this see the Clay Mathematics Institute website:

SolutionWe get a Klein bottle (discovered by Felix Klein in the late 19th century), which presents an important example of algebraic topology. It is a surface area that is not orientable, has no edges and which cannot be embedded into three-dimensional space. Nonetheless, it is possible to visualise it – a simple online search provides many images.

Timo de Wolff is a mathematician and a Professor at the Institute of Analysis and Algebra at the TU Braunschweig. He joined Die Junge Akademie in 2019.