My currently recommended combinatorics textbook by Pascal96 is a bit on the difficult side. Combinatorial: for competition reasons.(This is the same reason people’s first programming language is Python and not assembly.) I think this is an experience that is worth having, but it should not be the first experience one has. Intuitive: I reject the approach of some other instructors in which students start by proving basic results from first principles like the well-ordering principle, “all right angles are congruent”, etc.I believe this is important because I want to develop a student’s intuition, rather than try to teach them to work against it. For example, the statement that a tree always has one fewer edge than vertex is not obvious at first, so when one sees the proof it gives an idea. Substantial: the results one proves as practice should feel interesting.I have held a long belief that these are emphatically not the right way to start proofs, because in practice when one really does proofs, one is usually not thinking too much about the axioms of set theory. Pragmaticism: the textbook should not start with foundational issues like logical quantifiers or set theory.I specifically wanted to have the following requirements: There was a while I tried to look around to find an introduction-to-proofs textbook that I liked. At least let people get familiar with some mathematical objects, and learn what to expect from them, before you start formalizing everything. But that place is not a student’s first introduction to mathematical argument. Now there is a place for formal proof in mathematics, no question. RequirementsĬalling into question the obvious, by insisting that it be “rigorously proved”, is to say to a student, “Your feelings and ideas are suspect. For the contest kiddos out there, it basically amounts to saying “read the official solutions to any competition”.īut I think I can do better. There are a lot of students who ask me a question isomorphic to: I want to post it here to solicit opinions from the general community before investing a lot of time into the actual writing. This is a pitch for a new text that I’m thinking of writing.