Sorry, I’m a former high school mathematics teacher. Doing logic problems involving numbers doesn’t make them math. Logic problems like these require deductive reasoning skills, not mathematical ability. Deductive reasoning is a discipline of logic, which is taught in university philosophy departments, consistent with its origins with Aristotle.
You are referring to mathematical induction, which crosses over into deductive reasoning, but doesn’t encompass it.
No, I’m not referring to mathematical induction (which is a particular method of proof applicable to the natural numbers, or more generally well-ordered sets), I am referring to what mathematicians actually do. Almost all published mathematical research consists of proofs and general constructions. Very little involves calculations, which are usually the routine working out of instances of things mathematicians did in general, and are generally done by non-mathematicians, or even machines. My old grad school professor was deliberately using the asymmetry in the natural language use of “is” when he said, “Mathematics is proofs,” rather than “,Proofs are mathematics,” and he meant what he said.
BTW we mathematicians don’t think much of Aristotle — he got quantification over empty families wrong, and gave a mathematical description of physics that he could have debunked by watching his student Alexander’s soldiers loosing a volley of arrows or sling stones.
Evidently you must not have taught the right math courses at your school.
Back in the day, we has a whole year of number-free mathematics in high school called “plane geometry” in which we did proofs from Euclid’s postulates and compass-and-straight-edge constructions. It was actually much more like what mathematicians actually do than the dry rote derivation of simultaneous solutions to systems of linear equations, finding roots of polynomials, or implementing the recursive algorithm that finds formulas of derivatives for functions expressed in terms of arithmetic operations and composition of elementary functions.
The fact that according to you mathematicians don’t do mathematics, suggests your notion of what mathematics is needs a bit of brushing up.