How to learn mathematics from the building blocks? 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.
 A: 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.
A: 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.
