Learn mathematics
This page is a staged plan for learning mathematics from the beginning, written for someone whose last contact with the subject was school. It moves in a fixed order: first the vocabulary that mathematical writing assumes you already have, then the branches that group results into subjects, then the mathematicians who built them, and finally the primary texts. The order matters, because a proof in Euclid or Hilbert is unreadable without the words for what it is doing, and those words are easier to hold once you know which branch they came from. Nothing here needs more than arithmetic and school algebra to start. Every concept, branch, and mathematician carries its sources, mostly English Wikipedia, the MacTutor History of Mathematics archive at St Andrews, and the Stanford Encyclopedia of Philosophy for questions about foundations, so any claim here can be checked and followed further.
Stage 1. Learn the vocabulary
Finish this stage able to state what each of the eighteen terms means in your own words, and to give one example and one non-example of each.
Take the concepts in the order listed. The first four are about how mathematics argues, the next six are its basic objects, and the rest name specific structures. Read the definition here, then the linked article for that term, then close both and write the definition out from memory with an example you chose yourself. Two of these terms name results rather than objects, incompleteness and non-Euclidean geometry, and for those make sure you can say what the result rules out, not only what it says.
Stage 2. Map the branches
Finish this stage able to place each branch in time, say what it takes as given, and name the mathematicians who built it.
Read each branch summary and its core claims, then at least one cited source. The listed order is roughly chronological, and it is worth keeping, because the later branches are reactions to the earlier ones. Watch the pairs that argue with each other: Euclidean against non-Euclidean geometry, formalism against intuitionism. After each branch, write down which concepts from stage one it depends on. If you cannot name any, you have not finished reading it.
Stage 3. Meet the mathematicians
Finish this stage able to say, for each of the fifteen, what they proved, roughly when, and which branch the work belongs to.
Read each overview and note the work named as the starting point. Then read the MacTutor biography cited for that person; those entries are short and give the mathematics rather than only the life. Keep one page with fifteen lines on it, one result per mathematician, and rewrite that page from memory when you reach the end of the list.
Stage 4. Go to the sources
Finish this stage having read a primary mathematical text rather than a description of one.
Start with Euclid. Book I of the Elements is the shortest route into what a proof actually looks like, and John Casey's edition of the first six books is free on Project Gutenberg. After that there are three reasonable next steps, all free: the Archimedes treatise for ancient work on curved figures, Hilbert's Foundations of Geometry (1899) to see Euclid rebuilt to modern standards, and Poincaré's Science and Hypothesis for a working mathematician writing about mathematics for a general reader. Newton's Principia and Gauss on curved surfaces are harder and worth attempting only after the first three. Read with paper next to you and work each proof through yourself; following an argument with your eyes is not reading mathematics.
