Neckpinch — Journalhttps://neckpinch.com/journal/I'm teaching myself the mathematics behind Perelman's proof, in public, one day at a time. Thu, 24 Sep 2026 06:48:01 GMT Back to basics: Tao and the Peano axiomshttps://neckpinch.com/journal/tao-peano-axioms/https://neckpinch.com/journal/tao-peano-axioms/Wed, 23 Sep 2026 12:00:00 GMTStage 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 donehttps://neckpinch.com/journal/stage-one-done/https://neckpinch.com/journal/stage-one-done/Fri, 18 Sep 2026 12:00:00 GMTWhat 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 curvaturehttps://neckpinch.com/journal/spectral-theorem/https://neckpinch.com/journal/spectral-theorem/Sun, 13 Sep 2026 12:00:00 GMTSymmetric operators have orthonormal eigenbases. I think this is why people can talk about curvature as a list of numbers. Inner products put geometry back inhttps://neckpinch.com/journal/inner-products-gram-schmidt/https://neckpinch.com/journal/inner-products-gram-schmidt/Tue, 08 Sep 2026 12:00:00 GMTLength 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 turnhttps://neckpinch.com/journal/eigenvectors/https://neckpinch.com/journal/eigenvectors/Wed, 02 Sep 2026 12:00:00 GMTAxler's coordinate-free treatment of eigenvalues, and why every complex operator has at least one. Determinants finally mean somethinghttps://neckpinch.com/journal/determinants-as-volume/https://neckpinch.com/journal/determinants-as-volume/Thu, 27 Aug 2026 12:00:00 GMTI 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 bookkeepinghttps://neckpinch.com/journal/linear-maps-rank-nullity/https://neckpinch.com/journal/linear-maps-rank-nullity/Fri, 21 Aug 2026 12:00:00 GMTRank–nullity in Axler's language, and the realisation that a matrix is just a map wearing a particular basis. Vector spaces without coordinateshttps://neckpinch.com/journal/vector-spaces-without-coordinates/https://neckpinch.com/journal/vector-spaces-without-coordinates/Sun, 16 Aug 2026 12:00:00 GMTStarting 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 namehttps://neckpinch.com/journal/integrals-and-ftc/https://neckpinch.com/journal/integrals-and-ftc/Mon, 10 Aug 2026 12:00:00 GMTDefining 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 likehttps://neckpinch.com/journal/mean-value-theorem/https://neckpinch.com/journal/mean-value-theorem/Tue, 04 Aug 2026 12:00:00 GMTA 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 approximationhttps://neckpinch.com/journal/derivative-as-linear-approximation/https://neckpinch.com/journal/derivative-as-linear-approximation/Thu, 30 Jul 2026 12:00:00 GMTI 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 havinghttps://neckpinch.com/journal/least-upper-bounds/https://neckpinch.com/journal/least-upper-bounds/Fri, 24 Jul 2026 12:00:00 GMTCompleteness 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 ithttps://neckpinch.com/journal/continuity-and-ivt/https://neckpinch.com/journal/continuity-and-ivt/Sun, 19 Jul 2026 12:00:00 GMTThe intermediate value theorem says what your hand already knows. Proving it needs something deeper than intuition. Epsilon–delta, second attempt: it clickedhttps://neckpinch.com/journal/epsilon-delta-clicked/https://neckpinch.com/journal/epsilon-delta-clicked/Tue, 14 Jul 2026 12:00:00 GMTYes, 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)https://neckpinch.com/journal/epsilon-delta-first-attempt/https://neckpinch.com/journal/epsilon-delta-first-attempt/Thu, 09 Jul 2026 12:00:00 GMTI can recite the definition of a limit. I cannot yet use it. Writing down exactly where I get stuck. Functions aren't formulashttps://neckpinch.com/journal/functions-are-not-formulas/https://neckpinch.com/journal/functions-are-not-formulas/Fri, 03 Jul 2026 12:00:00 GMTI'd carried a wrong definition of "function" around for fifteen years without noticing. What a proof is supposed to look likehttps://neckpinch.com/journal/what-a-proof-looks-like/https://neckpinch.com/journal/what-a-proof-looks-like/Sun, 28 Jun 2026 12:00:00 GMTSpivak'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 makehttps://neckpinch.com/journal/day-one/https://neckpinch.com/journal/day-one/Wed, 24 Jun 2026 12:00:00 GMTI read about the Poincaré conjecture, couldn't stop thinking about it, and decided to learn the mathematics instead of admiring it from outside.