A language is a set of symbols that its users agree to apply to specific objects or concepts for the purpose of exchanging information. How does that apply to english and not mathematics? What criteria does english, or cantonese, meet that mathematics doesn't?
gmalivuk wrote:Sure, *after* using a real language to tell you what all the symbols I was using referred to in the cat and dog example.
At some point, someone used "real" language to explain to you what the symbols "cat", "dog" and "chased" mean. That's just translation. If you learn french, someone will explain to you what the french words for cat and dog are. The only reason you see french as a language and not mathematics is that the grammar of french is a lot closer to that of english.
Take japanese for an example - they drop a lot of words that we consider necessary for a "proper" sentence. "Go shop" isn't a sentence in english but it would be in japanese as they can infer the subject of the verb from who says it and when. Similarly, anyone looking at a mathematical description of "the cat chases the dog" would be able to infer what they needed to in order to understand what is going on.
This should be a lot more detailed but it's just an example to illustrate a point:
That tells a mathematician that something called "cat" is going in the same direction as something called "dog" but it's not going as fast. Add a couple more bits and pieces and you'd have a cat chasing a dog.