> 'Theoretical Mathematician'

what does that even mean ?

My first degree was in "Theoretical Math". I took classes in subjects like Number Theory, Topology, Analysis, Non-commutative Ring Theory. The subjects were about Math not how to do Math to solve other problems. We studied proofs not applying math to solve word problems.

As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math. An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.

Usually this is called pure mathematics, as opposed to applied mathematics.

On the other hand, applied mathematics might also involve some pretty technical "theoretical maths", which though being applied, can be studied for its own interest without the application.

So in some sense, "applied maths" sometimes means maths that is applied, rather than maths that can be applied. For that reason it perhaps makes sense to instead use the term "theoretical maths", though I am not sure if that is standard terminology or not.

In math, this means that we've established that he exists, but we haven't found him yet. Last I checked, they established a lower bound on his street address of 6325.