Mathematical Metaphors

BLOG: Heidelberg Laureate Forum

Laureates of mathematics and computer science meet the next generation
Heidelberg Laureate Forum

As part of the 8th HLF, we were treated to a panel discussion titled “Scientists Get Creative to Engage the Public with Science”, hosted by science writer and mathematician Susan D’Agostino, and featuring ACM A.M. Turing Award laureate and Chief Internet Evangelist at Google, Vint Cerf, Quanta Magazine’s Bill Andrews, statistician and award-winning mathematics communicator Talithia Williams, and TUM professor of geometry and visualization Jürgen Richter-Gebert.

The panel discussed many aspects of communication of science, summarised well in this post by fellow HLF blogger Andrei Mihai, and one topic the panel touched on was the importance of metaphors in communicating mathematics. Maths is an abstract subject which often involves visualising and simplifying complex structures, ideas and concepts, and metaphors can be a powerful tool in communication.

Getting Metaphorical

“I’ve found it very important to find common ground with the people I’m trying to communicate with”, says Vint Cerf, asking himself the question “Do they have any experience that I can take advantage of to help them understand something that’s very abstract?” Bill Andrews agrees; “[At Quanta], for us metaphor is really almost the bread & butter. […] Being able to find those translations from this highly abstract thought process into an average reader’s experience – we really care about that.” For the stories to make sense and resonate, they need to be understandable.

But other members of the panel were more cautious – Jürgen Richter-Gebert argued that metaphors can be dangerous, as they can be easily misunderstood. “In my work I do much more with experiments and true visualisations – I want to show people what is really going on; [… e.g.] creating a computational simulation of what you want to talk about, so people can get a really hands-on experience, and this somehow plays the role of a metaphor.”

He gave the example of an interactive experiment modelling COVID, which you could use to simulate distancing and experiment with different scenarios. Bill agreed with this cautious approach – “We are very careful not to get things wrong. If there’s a metaphor that’s misleading, we have fact-checkers that run things by our sources – we really want to make sure that our readers don’t get the wrong impression about anything.”

All the panellists agreed that for a metaphor to be good, it needs to involve something that’s familiar and everyday. Talithia gave the example of explaining the concept of stock options via the idea of a coupon you receive as a rain check, if you go to the store and they’re out of milk. The coupon guarantees you milk at the price of $2. If you go back to the store next week and the milk has gone up to $2.50, your coupon is great – but if you can just buy milk for $1.50, then it’s less useful. The question then is, how much could you sell this coupon for?

Talithia explains, “I’m always trying to ground my examples in personal experience – sharing things that people can resonate with – experiences they may have had, or that they know people who have had, to sort of pull them in to the situation.” It’s often the case that mathematical ideas can be explained using everyday objects, or by analogy to systems or structures people have already taken time to familiarise themselves with.

Some examples

One example is modular arithmetic – addition modulo some whole number n, taking only the remainder on division by n – underlies the theory of finite fields, which ties into coding theory and cryptography, and problems in number theory, including 2016 Abel Prize Laureate Andrew Wiles’ proof of Fermat’s Last Theorem. But who can imagine teaching a basic course on modular arithmetic without making reference to a clock face, and the idea of time wrapping round in a circle – so four hours past 10am is 2pm, in the same way that 10 + 4 ≡ 2 mod 12.

Image by foto-retusz from Pixabay

These kinds of simple analogies are invaluable in communicating mathematical concepts and some are so ubiquitous they find their way into undergraduate maths lecture halls and classrooms around the world. You may have learned about the process of mathematical induction by visualising a row of standing dominoes, falling one after the other to complete a proof. And if you’ve ever studied vector fields, you too may have pictured a surface covered in thick hairs, waving in the breeze as they point in different directions.

One lovely example I saw recently involved shoes and socks – if you’ve ever wrestled with explaining the composition of two functions together, you might have had students confused by the thought that inverting the composition of two functions means reversing the order of the functions. 

\[ (f \circ g)(x) = f(g(x)) \quad (f \circ g)^{-1} (x) = g^{-1}(f^{-1}(x)) \]

This kind of order reversal occurs in other areas of mathematics too – in composing two elements of a group, or applying operations to a matrix. But shoes and socks allow you to make sense of it – you put your socks on first, then put your shoes on over the top; but if you’re taking them off, you have to take your shoes off before you can take off your socks!

Metaphors in mathematics are sometimes so subtle they don’t even feel like a metaphor any more – for example, the set of real numbers is really just an uncountably infinite collection of objects with a defined ordering that allows us to know when one is greater than the other. But from the very beginning of nursery school, numbers are presented as existing on a line, with bigger numbers at one end and smaller numbers at the other. The real line is an incredibly powerful tool, and even when you reach higher levels of maths education and start considering open intervals on it, and proving that between every two fractions you can find infinitely many others, you’re still playing with that same straight line that was stuck on the wall in nursery class.

In fact, a lot of the concepts and structures in mathematics can almost work as metaphors in reverse – for example, graph theory allows us to condense a real-world structure, which can be quite complicated and contain lots of irrelevant information, into a simple network of dots and lines – and this will make it easier to understand, but while still encapsulating the nature of the object you’re studying.

There have also been some really effective metaphors used to encapsulate the experience of doing mathematics. Famously Andrew Wiles said:

“Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…”

But perhaps my favourite metaphor for the process of being a mathematician was encapsulated in this quote from 2014 Fields Medalist Maryam Mirzakhani:

“There are times when I feel like I’m in a big forest and don’t know where I’m going. But then somehow I come to the top of a hill and can see everything more clearly. When that happens, it’s really exciting.”

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is a mathematician based in Manchester, who gives talks and workshops on different areas of maths. She finished her PhD in 2011, and since then has talked about maths in schools, at science festivals, on BBC radio, at music festivals, as part of theatre shows and on the internet. Katie writes blog posts and editorials for The Aperiodical, a semi-regular maths news site.


  1. A metaphor in reverse
    An analyst wants to buy milk. But milk is sold out. Instead, the shopkeeper gives the analyst a sheet of paper with $ 2 scrawled on it. And the shopkeeper tells the analyst, Think of it as a stock option

  2. Die Mathematik ist wie ein Wald, der künstlich geschaffen ist, aber doch weitgehend unerforscht, es werden immer wieder Wege gefunden, was nicht leicht fällt, denn in diesem Wald ist es recht dunkel.
    Gefundene Wege haben manchmal, also nicht immer, einen Mehrwert, Anwendungen meinend, gesucht wird in diesem Wald aus Interesse am eigenen Fortkommen, die Mathematik kann als Wald der Fähigkeitslehre verstanden werden.

    Manchmal brechen Teile dieses Waldes zusammen, Wege zu finden wird dann unmöglich, was daran liegt, dass dieser Wald auf Grund verschiedener Axiomatiken entstanden ist bzw. gebildet worden ist, die Widersprüchlichkeit erzeugen kann, dann gilt es die Axiomatiken anzupassen, so dass der Wald dann verändert bereit steht.

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