The definition of mathematical truth is confusing.
Let P and Q denote the statements 'I'm a man' and '$1<x<2$', respectively.
P is truly (naturally) true because it is really true.
In natural language, we cannot determine whether Q is true or false. However in math, as far as I understand, Q has a truth set that is not empty; therefore it is mathematically true.
That there are two ways to determine whether something is true or false confuses me.
On
As a helpful addition, you may imagine your propositional variable to be a function from elements to truth.
In the case of any propositional variable with no free variables, we have a constant function, which (barring concerns of indexicality), always maps to a true or false (or whatever sort of truth values you have in your domain).
in the case of a propositional variable with n free variables, we obtain a n-ary function, such that, when substituted, maps to true or false. your Q for example maps 1.5 to true and 1 to false.
Now you can view a truth set as the preimage of "true" under the induced function. So these two methods are really instances of a more general method.
Note that Q is not mathematically true (or that your author is using a very strange definition). Rather Q(1.5) might be true, or Q(1) might be false.
In other words, P is synthetically true.
On the other hand, $Q$ is not a statement, and has no definite truth value.
When you say that $Q$ has a truth set, you mean that it is satisfied by every real number strictly between $1$ and $2.$ However, this does not mean that $Q$ is mathematically true: