| I looked up Bourbaki, my first go was at Algebra (chapters 1 to 3). Seems pretty decent, a bunch of things are obvious (has an introductory section which describes some of the motivation and historical context. Then starts by defining a law of composition using a function from E x E -> E: all pretty obvious. It even uses the common operators + and . (or no sign) to indicate addition and multiplication, all of which are intuitively clear and easy to make a parallel with what is already familiar. It even explicitely brings up a law of composition not everywhere defined on E for anyone who has not caught that composition needs to be a function on E x E -> E, or rather that it works for all values from E x E. For instance, subtraction on natural numbers is not a law of composition according to this definition. And straight up on the first page, there are examples of what compositions are available on the set of natural numbers and on subsets of sets. I am sure it gets hairer as you go along, but this is roughly the type of books I've used decades ago while studying mathematics, and roughly the type of books I enjoyed when properly interested and motivated. It's only obvious that one needs to go through the Set Theory first as they rely on the terminology introduced there for precise handling of whatever comes here (I still remember most of the terminology, but I don't trust my memory to get all the specifics right). It is not completely devoid of context and historical perspective, though it presents it in a slightly backwards way (introduction is clear to highlight how parts between asterisks are not necessary for purely logical reading of the text). Again, and I've said this before, formal mathematics is hard mostly because you have to memorize so much of the new language, and you can't really grasp the context without having grasped the context for what you are building on (eg. if you don't have understanding of functions/mappings, tough luck getting to the grips of algebraic structures). |
I enjoy that it starts off with defining symbols of a theory, and then assemblies, which is the first I hear of the term, but I can already feel what assemblies will amount to before the example given of an assembly in Theory of Sets — even if the text warns that "the meaning of this expression will become clear as the chapter progresses."
And that's exactly my point. I can enjoy learning from a text like this. I know I am in the minority (I was in the minority in Mathematical Gymnasium and university studies who enjoyed it; I wasn't even remotely the best at solving math problems, because I did not enjoy them), but I am calling for people not to discount this approach for everybody.