I use to think that algebra was very fundamental but it now seems not the case. The study of functions (in general) and sets would be more fundamental. Just an observation. What do you people think?
You could but lets say for up to graduate level, algebra involves a special kind of function called a homomorphism and consists of sets with special properties such as binary operation.The moment you start to do that -- you could get right back to algebra. e.g. categories.
"Fundamental" is such an incredibly vague term, though....