I do a lot of mathematics, so I believe I have some qualifications to answer this question objectively but uniquely and to my own taste. I hear very similar arguments in either situation: if someone believes that mathematics is a language, it is because they believe that it is capable of expressing ideas and facts, it contains symbols that can be read (whether from one language or multiple), and it is able to describe many things; if someone believes that mathematics is not a language, it is because they believe it is incapable of expressing emotion, complex thoughts, and does not possess qualities of art or creativity. Both reasons deserve more questioning.

While both hold some ideas that are true to an extent, I believe that the inability to classify whether mathematics is a language is down to the fact that people do not know enough mathematics: the reasons for either argument are misinformed. I believe that mathematics is distinct from language in the sense that it describes something that is universally true, whether applied in the sense of physical descriptions of phenomena, or pure in the sense that abstraction is the only possible expression of ideas. Language typically develops from a certain form of culture: culture can be defined very loosely, where two friends can have a culture between them and a group of people might find a cultural affinity in collecting watches. In either case, there is almost always a cultural syntax that is commonly understood between members of the culture, but if this language is exposed to members of a different culture, many details might be lost in translation. This occurs on a multitude of levels, because likely a watch-enthusiast would be able to communicate with non-watch-enthusiasts if they are descriptive enough; a culture between two friends might be difficult to explain to members of other cultures, as often the relationship between two individuals is so rooted in experience and familiarity that to understand it would be to be one of the said friends.

Mathematics, however, is different. Yes, while mathematics does have a culture in the sense that educational institutions often contort the meaning of who does mathematics - whether it is white men at Oxford University in the 19th century who twiddle their thumbs and “think about equations”, or young people pushing new ideas and theories that no one has heard of or is willing to hear about - the ambiguity of mathematical culture is so confusing that it is impossible to differentiate “this mathematics” from “that mathematics”. Under every single misconception and opinion regarding mathematics in any environment is a truth: every single well-established fact is true no matter what cultural syntax is used to express it. Some people believe pure mathematics is unimportant, but even if that may be true, it exists beyond the purpose of objective mathematical syntax: it just doesn’t matter.

I believe the first point at which you can truly consider what mathematics is is when you encounter proof. Many a mathematician will know how variable the way in which a result can be proven truly is, and I can say that I have written proofs in many different ways. Considerations of “elegance” and “non-elegance” are subjective, but they absolutely play a crucial role in determining whether something is beautiful or not. There are even many types of syntax with which a proof can be described, whether it is strictly using logic or using analogy. In either case, whether a proof uses a particular system of logic or field-related syntax, what it fundamentally describes is true no matter what.

This brings me to the following. Does it even mean anything to question whether mathematics is a language? Mathematics is so different from any human or non-human language because it simply does not possess a universal cultural syntax. Rather, it is an amalgamation of the vocabulary from every single possible culture and it exists beyond self and other. Whether it is a language has no meaning: the people or things who - or that - do mathematics have their own interpretation of what mathematics is and how it can be expressed. Humanity is not developed enough to understand how different mathematics looks in different cultural syntaxes, but a good example that would seem distant for many anyway would be something like the difference between two different fields: say category theory and number theory. While these certainly have many connections (I will not describe them because I am not familiar enough of either field), on the surface level, one would never think that number theory - something that can be described fairly easily (it’s definitely not easy, but the “big picture” ideas are more easily accessible) - is related to the often archaic seeming diagrams and ideas of category theory. Yet it is eye-opening to realize how much number theory really does fit in the framework of category theory: structures in number theory have vivid relationships with other structures in category theory.

Ultimately, I believe that mathematics deserves to be separate from language; by analogy, history is often attributed to the humanities, but more often than not it behaves more like a social science, where arguments are shaped and deduced rather than discovered. Mathematics is just that in a different way: a study that is neither a language nor a science, but something else entirely.

(1 July, 2024)