See now, I'd argue that the language comes after the mathematics. For example, I walk to work each day; part of walking to work is trying to find the route that lets me lie in the longest.

Now, humans are pretty good at exploring and finding alternative routes between locations, and they also tend to locate the shortest route given enough time.

Trying to explain how this intuitive activity works necessitates the use of graph theory. The graph theory was something our brain had constructed in the background, but it wasn't entirely conscious. Trying to explain this in natural language would take pages, however...

Given a set V of street intersections, and a set E of streets connecting two intersections, and a set W of weights assigned to each E. I can calculate the shortest route by applying one of the pathfinding algorithms (which are expressed in this notation).

This explanation will cover any pathfinding problem, but it's not great at conveying what is a really happening. The language we must use gets in the way of conveying the mathematics that is going on.

We do need a language (telepathy not being on the menu), but that language is a separate entity from the mathematics itself.

There are "mathematical languages", but these are present to describe mathematics. There are mathematical theories of language, but again the language itself is not mathematics - its structure, however, has mathematical properties.

I suppose you could say "fire has the property of being hot, but it isn't hotness itself"? Language is used to communicate mathematics, but it is not mathematics itself.

Now, this is not to discount notational developments in easing communication - that's a great branch as you have to check your new language and its rules match the mathematics it tries to describe. However, again, it's important not to conflate the thing you are describing with the thing you are using to describe it!

So, let's say you write down the words "fire is a chain reaction between carbon and oxygen that produces heat". You've characterised fire yes, but is that sentence itself the fire?

Let's say you write down the equation describing this reaction so you can play with it and manipulate it. Is this fire, or just a convenient way to talk about it?

I'd argue neither of these are fire, and both will never completely describe a fire (though they come damn close).

You know, I'd never considered that, but yes that does indeed seem to be the case!