### The Maths Muse

**The Language of Mathematics -*** by Aled Walker*

*by Aled Walker*

How might mathematics and language be related? Galileo felt that, “Mathematics is the language with which God wrote the universe.” This is the view of mathematics’ dual nature, as both a descriptive tool and as a fundamental component of that which it describes: one thinks of written language, of its double-life in the hands of a great novelist, their literary style both serving their narrative and being inseparable from it. On the other side of the galaxy, and of the cultural spectrum, T'Pol, the Vulcan crew member on ‘Star Trek: Enterprise’, remarked when trying to communicate with the Kreetassans that, “Math is just another language.” This is the view of mathematics as a language in and of itself, possessing such internal structure as to function as a self-contained method of communication: one thinks of spoken language, of some finite, amorphous collection of components which can somehow be combined to encompass the whole of human reality. Yet there is a third connection between mathematics and language, one far more concrete and tangible, which is sometimes overlooked, by both Renaissance men and Vulcan astronauts alike.

Although mathematics deals with extremely abstract concepts, which might seem beyond the reach of the descriptive powers of the English language, mathematicians are human beings at heart, and in order to be able to describe to each other their concepts, ideas, and structures, they have had to formulate a vocabulary in which to do this. And like all language, the language of mathematics does not exist in a vacuum, but rather exists within its own rich context, of associated connotations, references, definitions, and allusions. Yet unlike other uses of language even the most basic elements of its vocabulary are not considered to be a part of our shared lexicon, to be a part of assumed cultural knowledge, or to be fair game for general cross-disciplinary conversation. This is true in the UK at least: József Pelikán used to quip that when in 1975 Szemerédi proved the remarkable theorem which now bears his name, everybody in Hungary stopped to think about it!

Consider the following instance: if one wishes to begin know what quantum mechanics is really about, to know the most basic concepts which lie behind the tedious familiarity of the Schrödinger’s cat thought-experiment, then one must first know what a ‘Hermitian form’ is. No problem: Hermitian forms are self-adjoint linear operators on a complex inner-product space. But that sentence contained four notions which will probably be completely unfamiliar to the general reader: ‘self-adjoint’, ‘linear operator’, ‘complex’ (which has a technical meaning here), and ‘inner-product space’, to say nothing of the esoteric use of the word ‘form’. All of these are pieces of vocabulary from the language of mathematics, and they don’t just exist in isolation, but within a wider context of terms: one thinks of ‘vector space’, ‘matrix’, ‘eigenvalue’, ‘diagonalisation’, ‘conjugate-transpose’, ‘algebraic completion’, etc. etc.

I would argue that, by eschewing mathematical language from its frame of reference, our broader culture is depriving itself of great pleasure, knowledge, interest, and humour. In this article I hope to elucidate the issue of mathematical language further, and, in the style of John Lanchester’s diverting text ‘How to Speak Money’, to define a few of the elements of the standard mathematical lexicon with which any self-respecting conversationalist should be familiar.

*Originally from Birmingham, Aled Walker is a 4th year DPhil student in Mathematics at Magdalen College Oxford, and a Stipendiary Lecturer in Pure Mathematics at Pembroke College Oxford. His research tries to apply the methods of Additive Combinatorics to classical problems in Analytic Number Theory, with particular emphasis on the prime numbers. In 2010 he represented the UK at the International Mathematical Olympiad, and from October 2018 he will be a Junior Research Fellow at Trinity College Cambridge, where he was a Senior Scholar from 2011 to 2014. His other interests include jazz piano, lieder singing, and light operetta.*