I dated a woman many years ago who had a masters degree in theoretical math. Now 25 years later I wondered what the hell was theoretical math? Thank you for helping me gain some understanding.

This is my understanding of the difference between pure and applied: one’s motivation.

As you say, language and therefore meaning comes prior to grammar, and when one speaks of “pure mathematicians” he/she means those who do mathematics for no reason other than to do so, whereas “applied mathematicians” are doing mathematics entirely for the sake of solving a problem *outside of “mathematics proper”*.

One difficulty is that we are attempting to create a two-party mathematical state, to use a political metaphor, whereas these categories do not actually partition the field.

I personally use the term “theoretical mathematics” to obscure the motivation; i.e., the theories (methods, axioms, &c.) underlying the final results. This is in some amount of opposition to especially computational mathematics, where one applies previously-discovered mathematical methods to uncover new results, often in distinct fields (which, as observed by the author, often have or appear to have a one-to-one mapping with the relevant mathematical structures).

It seems that such results sometimes lie outside of pure mathematics because they are particularities, whereas pure mathematics seeks general statements. Of course, a number of such results may then result in a general pattern and so feed back into pure mathematics.

Via science, engineering, “applications”, thought, or happenstance, the mathematical theoretician is motivated by all, but seeks the theoretical, mathematical foundation upon which the results may be “explained” or “proved”.