Mathematical language is becoming more and more pervasive. This phenomenon ranges from the mundane (imprints on T-shirts or mugs) to the more scientific (its use in reporting or in disciplines outside of mathematics) and even includes art in its span. This begs the question, why and how does it work? Or more poignantly: What is the form and function of mathematical language inside and outside its community of speakers?
Mathematical text itself is highly stylized. Apart from its most abstract form of symbols and equations, the language contains diagrams (pictures) and stylized natural text, such as the standard composition structure: definition, lemma, theorem.
In the field of mathematics itself, the situation is not as homogenous as one might think. How much truth is contained in a proof by pictures is quite different in algebra versus geometry, and, historically, there is great variation in what is considered a proof—mainly how stylized the language should be. Being too relaxed can lead to foundational crises and questions like those Helmut Hofer is working out in symplectic geometry. An extreme position, which I call Frege’s dream, is also alive today with Vladimir Voevodsky and his colleagues through their endeavors to formalize language as much as possible to maximize verifiability. Some might argue that Bourbaki represented a golden age for striking a balance between the formal, the communal, and the communicable.
Wherever one falls on the spectrum of style and formality, the aim of mathematical language is to convey truth. Following Keats and the motto of the IAS, “Truth and Beauty” always appear conjointly. Where does beauty in mathematical language reside? Certainly, there is an aesthetic point of view from inside mathematics. Everybody has their own view of what beauty is, although there is certainly agreement on many singular items. One commonly accepted criterion for “elegant proofs” is simplicity. There is a bit more to it though for mathematicians. As Yuri Manin once put it, “Good proofs are proofs that make us wiser.” Yet simplicity as opposed to convoluted byzantine arguments acts as a beacon that is easily perceived, even by a more general audience. For example, E=mc2 is a famous formula with high recognition value and beauty in its simplicity. It is also an instance of physics using mathematical language. A venerable precedent is the Pythagorean theorem x2+y2=z2.
Both examples exhibit an almost universal appeal that takes hold well beyond the world of mathematics. They carry meaning beyond the community of people who know their exact context. There is something that remains, even if there is not a complete translation or transportation of content. The humorous illustrations of the discovery of these formulas play exactly on these aspects. So, what happens if you take mathematical language out of its community? I have used mathematical language inside different fields of mathematics (mostly about dialect), in research and conversations with physicists and chemical and materials engineers (requiring translation), but also with philosophers, the poet Oswald Egger, and, of course, just as a member of society.
Drawing on my personal experience, I posit that there is a certain scale for effectively communicating mathematical language and its associated truth and beauty. This type of communication is somewhat broken, however, in the sense that we can transmit ideas, but in order for them to be grasped, they have to be rediscovered or reimagined, much like in the translation of poetry. Here, beauty takes the lead as it is more facile to convey. Once beheld by the other party, it acts as a major indicator of truth and hence as a motivator for the regaining of the underlying truth through reconstruction. As Irving Lavin correctly pointed out, the positive result is the tree of knowledge growing between truth and beauty.
A good example, which contains all previously mentioned ingredients, is given in a cartoon by Robbert Dijkgraaf, IAS Director and Leon Levy Professor, which appeared on the cover of Quantum Fields and Strings: A Course for Mathematicians (American Mathematical Society, 1999), drawn from a special year-long program held at the Institute. It depicts the ongoing interdisciplinary conversations between physicists and mathematicians through an exchange of text and language. In both cases, it is necessary to reunderstand what is meant.
There is a strong gradient for peoples’ willingness to listen to abstract mathematics and their tolerance for it. Depending on their degree of familiarity with mathematical abstraction and symbols, this final step of decoding is what makes the task of communicating mathematical concepts or modes of thinking arduous. It also presents a hard problem for mathematicians, as we usually want to tell the full truth and this can be lengthy. The self-proclaimed aim of a mathematical text is to be universally understandable, although this requires a very strong commitment on the part of the reader.
It stands to reason that more pictures and natural language would be beneficial, but this bears some risk. Consider what happens when mathematical language is entirely removed from mathematics, such as in recent artworks by Bernar Venet, involving commutative diagrams. Without context and depending on the state of the diagram’s alteration by the artist, some of the truth is hard to extract, but it nevertheless illuminates the art. There are other artworks that exhibit a similar underlying beauty, such as sculptures of ruled surfaces, the Klein bottle, the gyroid surface, or neon art by Mario Merz based on the Fibonacci sequence, to name only a few.
One interesting aspect of some of these examples is that they incorporate more modern mathematics. For example, Oswald Egger was intrigued by the picture “a hole through a hole in a hole” and the Alexander horned sphere, which both deal with topology. He translated the beauty of these and other concepts into poetry. To make sure that truth remained a “golden thread,” Egger engaged me in discussions and took great efforts to understand the topological background.
The end product of mathematical activity is stark naked and condensed like a gem; it can be easy to forget that this is inherently a human activity and, although intentionally pushed to the background, it takes place within an individual’s thoughts and language. The thought-provoking titles above exemplify this, and in this light, it is not surprising that many of the technical terms of mathematics are shared with other disciplines, such as category, connection, and topos.
This is not a totally moot point; it provides a common ground from which to start a conversation. Historically, this is deeply rooted from Plato to Kant to Hegel to Cassirer, etc. Well-known examples are furnished by the Platonic solids and their role in Timaeus and the solution of a cubic equation in the form of a poem by Tartaglia in 1539. The poem is hardly readable today, and for us formulas would be easier. At the time, it was intended for a much greater audience of the learned society or even for the general public at market day contests. It is interesting to note that the full standard model Lagrangian is roughly the same length as the poem.
The following points may serve as a short summary. Mathematics strives toward truth, for which its language is designed. It can be made use of by exporting it to other disciplines or contexts. Reciprocally, mathematicians and mathematics gain a greater context. The language itself is complex and requires considerable effort, and direct application is difficult. But beauty, carefully communicated through illustrations, can help intuition and be a guide. This endeavor is, however, a worthy investment in truth and beauty. It will be great to see what modern mathematics can do and what the impact of its language will be on future society.