I know the original question title "what is the foundation of mathematics really?" seems pretty bad, or extremely bad, since it's really a huge question to answer. And I totally accept the fact that someone is going to disapprove this question.
But I have to ask this question anyway, the curiosity, or more kind of like pain, is killing me since it really has been bothering me for a very long period of time.
What is the foundation of mathematics really? Or the other way to state this question is like how those people lived in the past studied mathematics really?
I mean come on, I know nowadays people start to learn math from their elementary school doing simple things like addition or subtraction that sort of Arithmetics stuff, and as they move to middle school or high school, they start to learn geometry, or pre-calculus.
But in my perspective, these contents, or this path of learning math, is purely just a highly condensed abstraction that those people in education field designed that way. I have high confidence to assert that this can't be the way how people in the past learned math. By saying people in the past, I am really talking about peole in the past, like two or three hundred years ago. Moreover, to the very beginning of human kind.
I have done some researches on my own, and it seems like the foundation of modern mathematics is Euclid's elements, this reference is pretty much the one that I can find that looks like the very beginning of math to me. I am wondering if someone can recommend some references like Euclid's elements this sort of "very-beginning-of-math" to me.
Sincerely appreciated.
Best Answer
Victor Katz's A History of Mathematics does a pretty good job at tracing how some of our modern notions were developed, and sometimes how these things were conceptualized at the time and place. He gets a bit technical at times, being in an awkward spot between writing a history book and writing mathematics. But overall, I think it provides some of the context you're looking for, starting with Mesopotamian mathematics.
Of course, it's difficult to say definitively that such-and-such is definitively how they were thinking in comparison to now. So often enough the best we can do is instead describe how mathematics was used and taught. Lots of mathematics in the ancient world isn't based on proof but on algorithm or learned by example, where it seems expected that you follow and can generalize. Things are often stated in physical terms and appeal to intuition, being related to constructions, farm yields, and other things that ancient civilizations understandably prioritized.
Katz also goes into later mathematics in the later chapters, at first focusing on the development of calculus, but also touching on later developments and understandings of algebra, complex analysis, geometry, and a bit of logic and arithmetic. Part four of the book, for example, has chapters called "Analysis in the Eighteenth Century", "Algebra and Number theory in the Nineteenth Century", and "Aspects of the Twentieth Century and Beyond". The book more or less tries to follow a chronological telling, but obviously this isn't completely possible with things happening concurrently.