Mathematics might be described as (or more properly can be said to make use of) a formal language
, and is involved in studying the logically resulting structures, but as such does not itself include semantics. And so if a communication system cannot actually say things about the external world, I don't see how you can consider it a language in the sense that we talk about languages in this forum.
And when someone uses expressions like "the language of mathematics", they are typically referring to a combination of this formal language and mathematical jargon that is part of the actual human language being used to discuss the math in question.
I'm confused about your definition of semantics. I'm also not sure what your definition of saying things about the external world is, or why it is necessary for a language to have semantics. I don't think that the "language of math" is the same as mathematical jargon.
Let's say that I wanted to prove to you that every number divisible by four is an even number. I can only really do it in the language of math. Number x
is divisible by four means that there exists some n
such that 4*n
. Because 2*2=4, we can substitute that for each x
divisible by four, there exists n
such that (2*2)*n
. Multiplication of natural numbers is associative, so equivalently 2*(2*n
) = x
. Natural numbers are closed under multiplication, so (2*n
) is a natural number, lets call it m
. By substitution, we have demonstrated that for all x
divisible by four, there exists some m
such that 2*m
. But this implies that x
is even by definition.
Similarly, I could say that a number divisible by four must be divisible by two, since four is divisible by two, and the product of two numbers is divisible by any number that either is divisible by. I could also say that two is a prime factor of four and thus of any number divisible by four. There's no difference between the long formal proof and any of these simpler appeals. In all of them, I'm using the language of math. I'm doing so in English, granted, but the underlying mathematical semantics would remain unchanged when translated to any other language.