Thinking out loud
Journal
Short notes, most days: the definition that finally made sense, the computation I got wrong twice, the thing I'm still chewing on. I write them for myself, but they're open in case they help you too. If I've made a mistake — and I will have — please tell me.
18 entries, from June 2026 to now.
Back to basics: Tao and the Peano axioms
Stage 2 begins by going further backwards than I expected — before the real numbers, before the integers, to what 0 and "next" mean.
Stage 1 is done
What quadratic forms taught me about metrics, and an honest account of what three months bought me.
The spectral theorem, and my first real glimpse of curvature
Symmetric operators have orthonormal eigenbases. I think this is why people can talk about curvature as a list of numbers.
Inner products put geometry back in
Length and angle aren't part of a vector space until you add them. That single realisation reframed what a metric is.
Eigenvectors are the directions a map refuses to turn
Axler's coordinate-free treatment of eigenvalues, and why every complex operator has at least one.
Determinants finally mean something
I could compute determinants at nineteen without knowing what they were. It's a volume scaling factor, and that one sentence fixes everything.
Linear maps come first; matrices are bookkeeping
Rank–nullity in Axler's language, and the realisation that a matrix is just a map wearing a particular basis.
Vector spaces without coordinates
Starting Axler. His whole approach is to stop thinking of vectors as lists of numbers, and it's already paying off.
Integrals, and why the fundamental theorem deserves the name
Defining the integral properly took Spivak a whole chapter before a single one got computed. Now I see why.
The mean value theorem does more work than it looks like
A theorem I'd dismissed as a technicality turns out to be how you get from local information to global conclusions.
The derivative is the best linear approximation
I learned derivatives as slopes and rules. Restating them as linear approximation changed how the whole subject looks.
The axiom that makes the real numbers worth having
Completeness is one sentence, and it's the reason calculus works on ℝ and fails on ℚ.
Continuity, and a theorem that feels obvious until you try to prove it
The intermediate value theorem says what your hand already knows. Proving it needs something deeper than intuition.
Epsilon–delta, second attempt: it clicked
Yes, you do the algebra backwards and present it forwards. Once I accepted that, my first non-trivial limit proof took twenty minutes.
Epsilon and delta, first attempt (a failure)
I can recite the definition of a limit. I cannot yet use it. Writing down exactly where I get stuck.
Functions aren't formulas
I'd carried a wrong definition of "function" around for fifteen years without noticing.
What a proof is supposed to look like
Spivak's first chapter is twelve properties of numbers and a lot of humility. My first real proof was the triangle inequality.
Day one, and a promise I'm probably not qualified to make
I read about the Poincaré conjecture, couldn't stop thinking about it, and decided to learn the mathematics instead of admiring it from outside.