In school, math progresses more or less in a straight line: arithmetic, algebra, geometry, trigonometry, calculus. In college you may do some more advanced calculus, or a few other things like discrete math or differential equations, but that’s generally it.
But suppose you want to dive deeper. What comes after calculus?
I’m hardly an expert, but from what I can tell, modern mathematics – the stuff that real mathematicians work on – consists of three parts: analysis, topology, and abstract algebra.
I’ve started learning about abstract algebra.
In elementary algebra, you take numbers and add/subtract/multiply/divide them together. In abstract algebra, you take a step back. You say “Instead of numbers, let’s use any elements from a set,” and “Instead of adding/subtracting/multiplying/dividing, let’s do any operation that takes in two and spits out one.”
With those and a few other simple rules, you’re on your way.
The key insight here is that elementary algebra isn’t the only algebra, it’s just an algebra. There are others axiom-based systems that turn out to be just as good.
For instance, in matrix algebra, the things you operate on are matrices instead of just numbers. And multiplication is non-commutative (that is, AB doesn’t necessarily equal BA). How do we handle such a strange situation?
And it isn’t just matrices. Strange new algebras pop up everywhere you look, operating on anything you can think of. Rather than trying to figure out each one individually, abstract algebra asks: what can we say about the structure of mathematics in general?
I got this book from Amazon, and I’m working through it now. Good stuff so far.
What kind of math interests you?