One huge advantage of the notation conventions in math is its concision. The current practice is, at its heart, that a we assign a single glyph to each concept.

One huge advantage of the notation conventions in math is its concision. The current practice is, at its heart, that a we assign a single glyph to each concept. The notation is tailored for mathematicians communicating to other mathematicians. The terseness lets you communicate, visualize, and think about abstract concepts very directly. Humans are really good at pattern matching. Giving your concepts very terse names maximizes the ability for this tool to find actually interesting patterns. Of course, the terse notation sacrifices approachability. If your intent is to communicate to non-mathematicians then it's a bad language. That said, mathematical notation is full of historical quirks and irregularities akin to natural languages. Ideally, we want a one-to-one map between concepts and symbols, but in practice there are cases where the map ends up being more like many-to-many. There are competing notations, overloaded symbols, and often one needs significant context to actually resolve the symbol to its concept. In a way, programming languages are an inversion of the above good and bad points. Progamming languages are very precise with their syntax, meaning that you don't need to consider author's intent to figure out what a piece of syntax does. Also, programming languages, by virtue of being executable, are vastly more amenable to empower tinkering and exploration. There is something deeply satisfying about typing in some code and getting an expected result, and when the result differs from expectations, then you found a bug in your mental model. This is much harder to do with pen and paper. That said, most programming languages and their standard conventions fall short by being relatively wordy and verbose. What is a good one-liner in math symbols often takes up a page of code or more, not including definitions. This severely hamstrings the ability to grok the big picture at a glance, and opting for "words" over "symbols" intruces noise that kills our natural visual pattern matching abilities. So if you take the above thesis to its conclusion, one would want executable math notation. I'm not sure such a thing exists, but the APL family certainly is a solid experiment in this respect.