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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.

Pangi
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    Pretty vague question, especially since we don't know your background necessarily. Perhaps introductory real analysis is what you're looking for? – MT_ Jul 05 '16 at 21:45
  • Discrete math is what mathematicians consider to be the lowest level of math. It includes things like the definitions of numbers and basic operators as well as proofing and logic. Is that what you mean? – user64742 Jul 05 '16 at 21:53
  • I am looking for profs starting from axioms to explain high school math basically. I imagine mathematics as an uni-directed graph which has axioms as the only nodes with edges only pointing outwards and I would like to see the big picture of it. – Pangi Jul 05 '16 at 22:02
  • I guess discrete maths could give me a good starting point if it is the lowest level of math. I guess I can expand from there. But where should I start reading about it? – Pangi Jul 05 '16 at 22:18
  • It should also be stated that mathematics at an academic level looks very different from high school mathematics. But hypothetically, learning Set Theory and then Abstract Algebra, Real Analysis and Classical Geometry is exactly the answer to what you are asking for. Be prepared that that might even take you a few years though. – Georg Lehner Jul 06 '16 at 11:42

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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.

Reese Johnston
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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.

jose_castro_arnaud
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