Math is a Language
Math is a language, but there are many characteristics of this language which distinguish it from a spoken language such as English. For one thing, math is extremely precise, whereas a spoken language is vague by design. If spoken languages were not vague, it would be impossible to communicate even the simplest idea’s without going into great detail.
The language of mathematics is useful for describing the relationship between abstract concepts; and the precision of this language is fundamental to it’s ability to describe these relationships. A good example of an abstract concept for this discussion are the prime numbers.
According to modern mathematics, 1 is not a prime number by definition. It is sometimes claimed that this definition is arbitrary.
Mathematics is a system of rules (axioms) made up by people, and as opposed to physics where truth is determined by the result of experiments, in math true statements are derived from a system of axioms by using the rules of logic. In math, some ways of defining the meaning of words are more useful than others, in the sense that making certain definitions allows one to prove certain theorems that would otherwise be out of reach. The example of 1 not being prime illustrates this principle, since otherwise The Fundamental Theorem of Arithmetic, that every natural number has a unique prime factorization, would be false.
In a spoken language, it really would make no difference if we were to interchange the meaning of any two words, for example the words glove and shoe. Words in a spoken language are more like a label that are commonly agreed upon.
In math, the definitions of words are very important. Over time, the notation of mathematics becomes more concise, and this evolution in conciseness directly corresponds to an evolution of our understanding of the relationship between various abstract concepts. For example, Issac Newton wrote all the proofs in his Mechanics using geometric pictures, whereas today the same results can be derived much more elegantly using algebraic notation.
It’s fair to say that physics is about the way things are, whereas mathematics is about the way things have to be. The language of mathematics is not simply a collection of arbitrary axioms and definitions, it represents the collective effort of numerous generations to create a language capable of helping our minds to grasp abstract truths that are eternal.
