How to learn mathematics from the building blocks?

Question

From where can I learn mathematics from the basic blocks up? I feel like I have a lot of holes in the mathematics that I know and I would like to see where all those concepts come from. I would like to see what are the ideas that are took from granted, as foundation, and which ideas are made from this foundation.

Answer 1

Depends on what building blocks you want to study.

From the axioms up, one can begin with set theory, either naive or Zermelo-Fraenkel. Geometry has an additional set of axioms, for describing geometric concepts - see Foundations of Geometry.

On a more accessible level, one can study:

• Number theory, that assumes several common properties about integers, and builds up from them.
• Real analysis, about properties of real numbers, and its sequences, series and functions; it is the formalization of Calculus.
• Abstract algebra, about mathematical structures: sets with interesting operations defined on them.
• Linear algebra, which formalizes notions like vectors and matrices.

You can use Wikipedia as a starting point - pay attention to the external links, for widening the search - and try searches like "book on (math field here)". There are many e-books out there.

Answer 2

If you want proofs from axioms, then Euclid's Elements is a classic example for geometry. For arithmetic/high-school algebra, there's Peano Arithmetic - I'm not sure what would be a good book to learn that from, but PA is a collection of axioms generally acknowledged to cover everything you'd care to know about the natural numbers at the high-school algebra. Depending on your level, though, the logic required for PA might be a little heavy.