Math as a Language

I've discussed about how mathematics as a form of art in one of past essays. Now, not only can you find mathematics in beauty, you can also find it in languages. Semantics, grammars, syntax, morphology, discourse, pragmatics, ... These are mainly discussed in an anthropology or linguistic class. However, I find semantics, syntax, and morphology to highly familiar in mathematics. Semantics in mathematics can simply just the idea and visualization of discrete math. Syntax in mathematics are the common operators that we see daily with =, <, >, or even the sigma function (that I don't know how to write in markdown). Finally, morphology is a core concept in different fields in mathematics like linear algebra, discrete math, or even abstract algebra. "Mathmatical morphology" is actually a pretty cool field about processing and analysis of image, using operators and functionals based on topological and geometrical concepts (yeah, i got that from wiki). Mathematic language also need to have structure like languages. You can see this idea as the grammar in mathematics. An equation need to have expressions on two sides with the "=" syntax in the middle. An equation can be false if the two expressions are not semantically return and point to the same value. Writing 1 = 1 and saying I am myself all return true.

This can link us to the idea of logical operators with /forall, /exists, /neg, /wedge, /vee. These can all be define to for all, exists at least one, not, and, or. They literally have identical semantics in both mathematics and languages in general. Consider A and B to be expressions or statements. By using these logical operators, you are literally creating sentences/statements in both mathematics and verbal languages. There're many more examples of this in mathematics but this is probably the easiest representation of how plain english and math can have the same semantics. However, I am not saying that any sentences in english can be converted to a true and reasonable mathematics statements. Sentence like "This statements is false" simply do not have a finite answer in math and basically an infinite paradox. This concept can be linked to Godel's incomplete theorem that you can research more on. Therefore, you can represent any math with english but definitely not vice versa, showing that Math's incompleteness and also english verbosity when it can describe impossible statements. This also makes plain languages not rigorous. I believe this is one of the reason why just plain english can not always be used for an LLM to make a rigorous and provable mathematical proofs or algorithms.

This essay is on one of the topic that I hardly have any expertise on but a good way for me to explain my thoughts on mathematics as a language and the weirdness of our good old english. English is really informal and proofs and programs are probably the most formal things I can think of. English is just not a good choice for the formal specifications of programs. That's why we need programming languages and proof assistants.