Notes, projects, records, and games from Deep0Thinking.https://deep0thinking.com/enhttps://deep0thinking.com/notes/a-guide-for-preparing-for-the-william-lowell-putnam-mathematical-competition/https://deep0thinking.com/notes/a-guide-for-preparing-for-the-william-lowell-putnam-mathematical-competition/Preparation ideas, references, and expectations for the William Lowell Putnam Mathematical Competition.Sat, 05 Sep 2026 00:00:00 GMTcompetitioncompetitiveguideintroductionlatexmathematical-competitionmathematicshttps://deep0thinking.com/notes/a-mathematical-exploration-of-nortons-dome-and-determinism-in-classical-mechanics/https://deep0thinking.com/notes/a-mathematical-exploration-of-nortons-dome-and-determinism-in-classical-mechanics/A mathematical look at Norton's dome, non-unique motion, and determinism in classical mechanics.Sat, 05 Sep 2026 00:00:00 GMTchineseclassical-mechanicsdifferential-equationintroductionlatexmathematicsparadoxphysicshttps://deep0thinking.com/notes/a-mathematical-exploration-of-the-virial-theorem/https://deep0thinking.com/notes/a-mathematical-exploration-of-the-virial-theorem/A derivation and interpretation of the virial theorem through the mathematics of classical mechanics.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsintroductionlatexmathematicsphysicshttps://deep0thinking.com/notes/a-simple-analogy-to-understand-terminal-shell-tty-and-console/https://deep0thinking.com/notes/a-simple-analogy-to-understand-terminal-shell-tty-and-console/A practical mental model for distinguishing terminals, shells, TTYs, and consoles.Sat, 05 Sep 2026 00:00:00 GMTchinesecomputer-scienceintroductionlinuxoperating-systemhttps://deep0thinking.com/notes/a-simple-physical-approach-to-understand-the-taylor-series/https://deep0thinking.com/notes/a-simple-physical-approach-to-understand-the-taylor-series/An intuitive route from motion with changing acceleration to the structure of a Taylor series.Sat, 05 Sep 2026 00:00:00 GMTguidemathematicsphysicstaylor-serieshttps://deep0thinking.com/notes/advice-for-new-python-learners/https://deep0thinking.com/notes/advice-for-new-python-learners/Practical advice about editors, IDEs, and learning habits for people beginning Python.Sat, 05 Sep 2026 00:00:00 GMTchineseexperienceguideideprogrammingpythontext-editorhttps://deep0thinking.com/notes/apex-legends-game-sense/https://deep0thinking.com/notes/apex-legends-game-sense/Notes on the mistakes I keep making in Apex and how I try to stop repeating them. This is only part of game sense, and only the part I have managed to name so far.Sat, 05 Sep 2026 00:00:00 GMTapex-legendsfpsbattle-royalemultiplayergamingguidehttps://deep0thinking.com/notes/apex-legends-settings-and-gear/https://deep0thinking.com/notes/apex-legends-settings-and-gear/The settings, keybinds, sensitivity, and gear I actually play on.Sat, 05 Sep 2026 00:00:00 GMTapex-legendsfpsbattle-royalemultiplayersettingssensitivitygeargaminghttps://deep0thinking.com/notes/competitive-mathematics-factorization/https://deep0thinking.com/notes/competitive-mathematics-factorization/A structured guide to polynomial factorization, identities, methods, and proofs for mathematical competitions.Sat, 05 Sep 2026 00:00:00 GMTcompetitioncompetitiveguidelatexmathematical-competitionmathematicsnoteshttps://deep0thinking.com/notes/could-the-speed-of-light-be-variable/https://deep0thinking.com/notes/could-the-speed-of-light-be-variable/What the constancy of light speed means, how it is tested, and where variable-speed ideas would have to differ.Sat, 05 Sep 2026 00:00:00 GMTexperimental-physicsmathematicsphysicsspecial-relativityspeed-of-lighthttps://deep0thinking.com/notes/cracking-the-logic-gates-game-turing-complete/https://deep0thinking.com/notes/cracking-the-logic-gates-game-turing-complete/Using mathematical logic and Python to reason through a logic-gate construction challenge in Turing Complete.Sat, 05 Sep 2026 00:00:00 GMTcodingcomputer-sciencegamingintroductionlatexlogiclogic-gatemathematical-logicmathematicsprogrammingpythonhttps://deep0thinking.com/notes/history-of-logic-01/https://deep0thinking.com/notes/history-of-logic-01/Part 0 introduces the series, its reading order, and routes through foundations, computation, formalization, and earlier logical traditions.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicfoundations-of-mathematicsformalizationhistory-of-philosophyinteractive-learninghttps://deep0thinking.com/notes/history-of-logic-02-boole-and-algebra/https://deep0thinking.com/notes/history-of-logic-02-boole-and-algebra/How an algebra of classes became a calculus of relations and quantifiers, through Boole, De Morgan, Peirce, and Schröder.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicalgebraic-logichistory-of-mathematicsboolean-algebrarelationsquantifiersinteractive-labshttps://deep0thinking.com/notes/history-of-logic-03-frege-and-quantifiers/https://deep0thinking.com/notes/history-of-logic-03-frege-and-quantifiers/Why variables, scope, and explicit inference changed the representation of mathematical proof, and how Frege's project differed from modern first-order logic.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicformal-languagefoundations-of-mathematicsquantifiersfirst-order-logiclogicismhigher-order-logicinteractive-labshttps://deep0thinking.com/notes/history-of-logic-04-rigor-infinity-and-axioms/https://deep0thinking.com/notes/history-of-logic-04-rigor-infinity-and-axioms/How analysis, infinite sets, arithmetic, and alternative geometries changed the questions mathematicians asked about foundations.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicfoundations-of-mathematicsset-theoryaxiomatizationinfinityreal-analysisnon-euclidean-geometryinteractive-labshttps://deep0thinking.com/notes/history-of-logic-05-paradoxes-and-foundations/https://deep0thinking.com/notes/history-of-logic-05-paradoxes-and-foundations/Russell's contradiction, Frege's Basic Law V, ramified types, predicativity, and axiomatic set theory as distinct responses to foundational problems.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicfoundations-of-mathematicsset-theorytype-theorylogicismparadoxesrussells-paradoxpredicativityaxiomatizationinteractive-labshttps://deep0thinking.com/notes/history-of-logic-06-hilberts-program/https://deep0thinking.com/notes/history-of-logic-06-hilberts-program/Formalization, finitary reasoning, and the attempt to justify infinitary mathematics through a mathematical analysis of finite derivations.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicproof-theoryconsistencyfoundations-of-mathematicshilberts-programfinitismepsilon-calculusformalizationhttps://deep0thinking.com/notes/history-of-logic-07-intuitionism-and-construction/https://deep0thinking.com/notes/history-of-logic-07-intuitionism-and-construction/Brouwer, Heyting, constructive evidence, realizability, and the different mathematical programs behind the word constructive.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicconstructivismproof-theoryintuitionistic-logicrealizabilityclassical-logichttps://deep0thinking.com/notes/history-of-logic-08-syntax-truth-and-completeness/https://deep0thinking.com/notes/history-of-logic-08-syntax-truth-and-completeness/The distinction between derivability and semantic consequence, from formal syntax and Tarski's semantics to Gödel's theorem, Henkin's construction, and nonstandard models.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicsemanticsmodel-theorycompletenessfirst-order-logictruthcompactnesslowenheim-skolemhttps://deep0thinking.com/notes/history-of-logic-09-godel-and-incompleteness/https://deep0thinking.com/notes/history-of-logic-09-godel-and-incompleteness/How arithmetic represents proofs, how diagonalization produces an undecidable sentence, and what the two incompleteness theorems establish.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicincompletenessproof-theoryfoundations-of-mathematicsgodels-theoremsarithmeticdiagonalizationconsistencyhttps://deep0thinking.com/notes/history-of-logic-10-computability/https://deep0thinking.com/notes/history-of-logic-10-computability/Church, Turing, Kleene, Post, and the mathematical analysis of algorithms, undecidability, relative computation, and Diophantine equations.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logiccomputabilityalgorithmsturing-machinesundecidabilityhalting-problemlambda-calculusinteractive-labshttps://deep0thinking.com/notes/history-of-logic-11-gentzen-and-proof-theory/https://deep0thinking.com/notes/history-of-logic-11-gentzen-and-proof-theory/Natural deduction, sequent calculus, cut elimination, ordinal analysis, and the continuing investigation of the strength and computational content of proofs.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicproof-theorynatural-deductionconsistencyreverse-mathematicssequent-calculuscut-eliminationordinal-analysisinteractive-labshttps://deep0thinking.com/notes/history-of-logic-12-model-theory/https://deep0thinking.com/notes/history-of-logic-12-model-theory/Definability, quantifier elimination, ultraproducts, nonstandard analysis, and classification as tools for understanding mathematical structures.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicmodel-theorysemanticsdefinabilityquantifier-eliminationultraproductsnonstandard-analysisstability-theoryhttps://deep0thinking.com/notes/history-of-logic-13-set-theory-and-independence/https://deep0thinking.com/notes/history-of-logic-13-set-theory-and-independence/Choice, the continuum hypothesis, Gödel's constructible universe, Cohen's forcing, and the continuing question of which axioms to adopt.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicset-theoryindependencefoundations-of-mathematicsforcingcontinuum-hypothesisaxiom-of-choiceconstructibilityhttps://deep0thinking.com/notes/history-of-logic-14-modal-logic/https://deep0thinking.com/notes/history-of-logic-14-modal-logic/How the study of implication developed into a family of logics whose operators express different kinds of necessity.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicmodal-logicsemanticskripke-semanticstemporal-logicepistemic-logicprovability-logicinteractive-labshttps://deep0thinking.com/notes/history-of-logic-15-alternative-logics/https://deep0thinking.com/notes/history-of-logic-15-alternative-logics/Four distinct reasons to reconsider classical inference, illustrated through truth values, relevant implication, inconsistent information, and structural rules.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicnonclassical-logicproof-theorylinear-logicmany-valued-logicrelevance-logicparaconsistent-logicstructural-ruleshttps://deep0thinking.com/notes/history-of-logic-16-lambda-calculus-and-curry-howard/https://deep0thinking.com/notes/history-of-logic-16-lambda-calculus-and-curry-howard/How substitution became a theory of computation, and how typed terms came to represent proofs with computational behavior.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logiclambda-calculustype-theorycurry-howardproof-theorycomputationnormalizationinteractive-labshttps://deep0thinking.com/notes/history-of-logic-17-dependent-type-theory/https://deep0thinking.com/notes/history-of-logic-17-dependent-type-theory/How types came to express specifications that depend on values, and why equality, induction, and universe rules became foundational decisions.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicdependent-typesfoundations-of-mathematicsproof-assistantstype-theoryidentity-typesuniversesinductive-typeshttps://deep0thinking.com/notes/history-of-logic-18-categorical-logic/https://deep0thinking.com/notes/history-of-logic-18-categorical-logic/How categories provided structural interpretations of proof, function types, quantifiers, and mathematical foundations.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logiccategorical-logicsemanticsfoundations-of-mathematicscategory-theoryadjunctionstopos-theorytype-theoryhttps://deep0thinking.com/notes/history-of-logic-19-functional-programming-and-semantics/https://deep0thinking.com/notes/history-of-logic-19-functional-programming-and-semantics/From Lisp and lambda-based language design to mathematical accounts of recursion, polymorphic type inference, and abstraction.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicfunctional-programmingprogramming-languagessemanticslambda-calculusdenotational-semanticstype-inferencedomain-theoryhttps://deep0thinking.com/notes/history-of-logic-20-automated-theorem-proving/https://deep0thinking.com/notes/history-of-logic-20-automated-theorem-proving/How proof procedures turn a mathematical problem into a search, and why representation, inference rules, and search strategy each matter.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicautomated-reasoningresolutiontheorem-provingunificationequalityproof-searchfirst-order-logichttps://deep0thinking.com/notes/history-of-logic-21-complexity-sat-and-smt/https://deep0thinking.com/notes/history-of-logic-21-complexity-sat-and-smt/Why decidable reasoning can be difficult, how conflict-driven solvers exploit structure, and how Boolean search cooperates with mathematical theories.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logiccomplexitysatsmtautomated-reasoningnp-completenessdpllcdcldecision-proceduresinteractive-labshttps://deep0thinking.com/notes/history-of-logic-22-logic-programming/https://deep0thinking.com/notes/history-of-logic-22-logic-programming/How proof search can compute answers, and why a program's logical consequences must be distinguished from the behavior of its execution strategy.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logiclogic-programmingprologautomated-reasoninghorn-clausessld-resolutiondatalogunificationhttps://deep0thinking.com/notes/history-of-logic-23-proof-assistants/https://deep0thinking.com/notes/history-of-logic-23-proof-assistants/The distinct histories of Automath, LCF, Mizar, inductive provers, dependent-type systems, Metamath, and Lean—and what each architecture asks us to trust.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicproof-assistantsformalizationfoundations-of-mathematicstype-theorytrusted-kernelleanlcfcoqhttps://deep0thinking.com/notes/history-of-logic-24-program-verification/https://deep0thinking.com/notes/history-of-logic-24-program-verification/How assertions became a logic of programs, and how proofs of loops, heap operations, compilers, and kernels depend on precise specifications.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicprogram-verificationhoare-logicseparation-logicformal-methodsloop-invariantsweakest-preconditionscompiler-correctnesshttps://deep0thinking.com/notes/history-of-logic-25-model-checking/https://deep0thinking.com/notes/history-of-logic-25-model-checking/How logics of time became tools for checking ongoing systems, and how symbolic representations and abstraction address the growth of possible behaviors.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicmodel-checkingtemporal-logicabstract-interpretationformal-methodssafety-and-livenesssymbolic-verificationabstractioninteractive-labshttps://deep0thinking.com/notes/history-of-logic-26-formalized-mathematics/https://deep0thinking.com/notes/history-of-logic-26-formalized-mathematics/How mechanically checked mathematics grew from individual texts and landmark proofs into shared infrastructure for further mathematical work.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicformalized-mathematicsmathematical-librariesproof-assistantsformalizationproof-reflectionmathlibleanhttps://deep0thinking.com/notes/history-of-logic-27-homotopy-type-theory/https://deep0thinking.com/notes/history-of-logic-27-homotopy-type-theory/How identity types acquired a geometric interpretation, why equivalence can induce identification, and how cubical theories address computation.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logichomotopy-type-theoryunivalencefoundations-of-mathematicstype-theorydependent-typesidentity-typescubical-type-theoryhttps://deep0thinking.com/notes/history-of-logic-28-learned-theorem-proving/https://deep0thinking.com/notes/history-of-logic-28-learned-theorem-proving/How statistical guidance works with formal checking, from premise selection to AlphaProof and research formalization, with careful distinctions about evidence and novelty.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmathematical-logicaitheorem-provingformalizationautomated-reasoningmachine-learningproof-searchautoformalizationhttps://deep0thinking.com/notes/history-of-logic-earlier-01-greek-logic/https://deep0thinking.com/notes/history-of-logic-earlier-01-greek-logic/Aristotelian demonstration, syllogistic inference, Stoic arguments, and the commentators and translators who shaped their later study.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicancient-logicaristotlestoicismhistory-of-philosophysyllogistic-logicpropositional-logicdemonstrationhttps://deep0thinking.com/notes/history-of-logic-earlier-02-arabic-islamic-jewish-logic/https://deep0thinking.com/notes/history-of-logic-earlier-02-arabic-islamic-jewish-logic/Translation, demonstration, Avicennan innovations, later teaching traditions, and the movement of logical ideas across Arabic, Hebrew, and Latin.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicarabic-logicavicennamedieval-logicislamic-philosophyjewish-philosophysyllogistic-logictranslationhttps://deep0thinking.com/notes/history-of-logic-earlier-03-medieval-latin-logic/https://deep0thinking.com/notes/history-of-logic-earlier-03-medieval-latin-logic/How medieval logicians analyzed the use of terms, the force of logical particles, valid consequences, and self-referential statements.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicmedieval-logicsemanticsparadoxessupposition-theorylogical-consequenceself-referencehistory-of-philosophyhttps://deep0thinking.com/notes/history-of-logic-earlier-04-indian-logic/https://deep0thinking.com/notes/history-of-logic-earlier-04-indian-logic/Nyāya, Buddhist theories of inference, and Navya-Nyāya's technical language, examined through the justification and failure of inferential signs.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicindian-logicnyayabuddhist-logicepistemologyinferencenavya-nyayahistory-of-philosophyhttps://deep0thinking.com/notes/history-of-logic-earlier-05-chinese-argument/https://deep0thinking.com/notes/history-of-logic-earlier-05-chinese-argument/Mohist standards of reasoning, the School of Names, the white-horse discussion, and Xunzi's account of naming and orderly discourse.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicchinese-philosophymohismargumentationphilosophy-of-languageanalogyschool-of-nameshistory-of-philosophyhttps://deep0thinking.com/notes/history-of-logic-earlier-06-before-boole/https://deep0thinking.com/notes/history-of-logic-earlier-06-before-boole/Early-modern projects for improving inquiry and calculation, followed by Bolzano's account of propositions, variation, consequence, and explanation.Sat, 05 Sep 2026 00:00:00 GMThistory-of-logicearly-modern-logicleibnizbolzanoinductionlogical-consequencesymbolic-logichistory-of-philosophyhttps://deep0thinking.com/notes/how-is-diandian-a-cat-my-family-saved-2-years-ago-now/https://deep0thinking.com/notes/how-is-diandian-a-cat-my-family-saved-2-years-ago-now/The story of the kitten my family rescued in February 2022, with a 2024 video update.Sat, 05 Sep 2026 00:00:00 GMTchinesediandianexperiencefamilylovepetvideomemoryhttps://deep0thinking.com/notes/how-separation-resolves-russells-paradox/https://deep0thinking.com/notes/how-separation-resolves-russells-paradox/Unrestricted comprehension is inconsistent. Separation replaces it, and the relative Russell set is a subset of its base set but never an element of it.Sat, 05 Sep 2026 00:00:00 GMTmathematical-logicset-theorylogicfoundationsparadoxproofmathematicshttps://deep0thinking.com/notes/i-switched-from-jekyll-to-wordpress/https://deep0thinking.com/notes/i-switched-from-jekyll-to-wordpress/Notes from the 2023 move from a Jekyll site on GitHub Pages to WordPress.Sat, 05 Sep 2026 00:00:00 GMTbackendblogdeveloperfrontendgithubsiteweb-developmentwebsitemilestonehttps://deep0thinking.com/notes/i-switched-from-wordpress-to-astro/https://deep0thinking.com/notes/i-switched-from-wordpress-to-astro/Notes from the move off WordPress to a static Astro site, what I gained, and what I gave up.Sat, 05 Sep 2026 00:00:00 GMTaiastroblogdesignmigrationsitestatic-siteweb-developmentwebsitemilestonehttps://deep0thinking.com/notes/intro-to-generalized-coordinates/https://deep0thinking.com/notes/intro-to-generalized-coordinates/An introduction to generalized coordinates and why they simplify constrained mechanical systems.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsintroductionlagrangian-mechanicsmathematicsphysicshttps://deep0thinking.com/notes/intro-to-git-and-github/https://deep0thinking.com/notes/intro-to-git-and-github/A bilingual guide to repositories, forks, clones, branches, commits, and pull requests, with a complete local workflow.Sat, 05 Sep 2026 00:00:00 GMTchinesecomputer-sciencegitgithubguideintroductionprogrammingvscodehttps://deep0thinking.com/notes/intro-to-the-colemak-keyboard-layout/https://deep0thinking.com/notes/intro-to-the-colemak-keyboard-layout/A concise introduction to Colemak, its layout choices, and the process of learning it.Sat, 05 Sep 2026 00:00:00 GMTcomputer-scienceintroductionkeyboard-layouttypinghttps://deep0thinking.com/notes/lorentz-transformations-and-general-relativity/https://deep0thinking.com/notes/lorentz-transformations-and-general-relativity/How local inertial frames connect Lorentz transformations in special relativity to curved spacetime.Sat, 05 Sep 2026 00:00:00 GMTdifferential-geometrygeneral-relativitymanifoldmathematicsmetric-spacephysicsspecial-relativityhttps://deep0thinking.com/notes/my-experience-of-u-s-visa-interview-in-guangzhou-china/https://deep0thinking.com/notes/my-experience-of-u-s-visa-interview-in-guangzhou-china/A first-person account of preparing for and completing a U.S. visa interview in Guangzhou.Sat, 05 Sep 2026 00:00:00 GMTchineseexperiencehttps://deep0thinking.com/notes/recommendations-for-rigorous-classical-mechanics-textbooks/https://deep0thinking.com/notes/recommendations-for-rigorous-classical-mechanics-textbooks/A reading list for approaching classical mechanics with greater mathematical rigor.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsexperienceformalismguidemathematicsphysicsrecommendationtextbookhttps://deep0thinking.com/notes/summation-by-parts-abel-transformation/https://deep0thinking.com/notes/summation-by-parts-abel-transformation/An introduction to summation by parts and its role as the discrete analogue of integration by parts.Sat, 05 Sep 2026 00:00:00 GMTalgebrachineseintroductionlatexmathematicshttps://deep0thinking.com/notes/using-raspberry-pi-to-build-a-us-vpn-server-for-my-family-in-china/https://deep0thinking.com/notes/using-raspberry-pi-to-build-a-us-vpn-server-for-my-family-in-china/A practical record of building a family VPN with Raspberry Pi, SSH, DNS, and Cloudflare.Sat, 05 Sep 2026 00:00:00 GMTcloudflarecodingcomputer-scienceddnsdnsgithubgreat-firewalllinuxnetworkprogrammingraspberry-piserversoftwaresshvideovpnhttps://deep0thinking.com/notes/why-are-generalized-coordinates-in-the-lagrangian-equation-not-considered-functions-of-time/https://deep0thinking.com/notes/why-are-generalized-coordinates-in-the-lagrangian-equation-not-considered-functions-of-time/How partial derivatives of a Lagrangian differ from differentiation along a time-dependent trajectory.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsintroductionlagrangian-mechanicslatexmathematicsphysicshttps://deep0thinking.com/notes/why-differentiating-both-sides-does-not-solve-equations/https://deep0thinking.com/notes/why-differentiating-both-sides-does-not-solve-equations/Squaring both sides keeps every original solution, but differentiating does not. The difference comes from the type of the function you apply.Sat, 05 Sep 2026 00:00:00 GMTalgebrachinesedifferential-equationmathematical-logicmathematicsproofhttps://deep0thinking.com/notes/why-friction-breaks-time-reversal-symmetry/https://deep0thinking.com/notes/why-friction-breaks-time-reversal-symmetry/How microscopic reversibility, probability, and coarse-graining produce macroscopic irreversibility.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsloschmidts-paradoxmathematicsparadoxphysicsprobabilitystatistical-mechanicsthermodynamicstime-reversal-symmetryhttps://deep0thinking.com/notes/why-linear-subspaces-contain-zero/https://deep0thinking.com/notes/why-linear-subspaces-contain-zero/Why nonemptiness and scalar closure force a subspace to contain the ambient zero vector.Sat, 05 Sep 2026 00:00:00 GMTlatexlinear-algebramathematicsmodern-algebrahttps://deep0thinking.com/notes/why-mathematical-logic-textbooks-look-circular/https://deep0thinking.com/notes/why-mathematical-logic-textbooks-look-circular/Standard textbooks use sets to define first order logic, then define set theory inside first order logic. Where that apparent circularity resolves, and what the metatheory consists of.Sat, 05 Sep 2026 00:00:00 GMTmathematical-logiclogicfoundationsmetamathematicsformalismleantextbookmathematicshttps://deep0thinking.com/notes/why-phasors-are-used-in-circuit-analysis/https://deep0thinking.com/notes/why-phasors-are-used-in-circuit-analysis/How phasors turn sinusoidal circuit equations into algebra, and what their components represent physically.Sat, 05 Sep 2026 00:00:00 GMTalgebracircuit-analysiscircuitsdifferential-equationelectromagnetismlinear-algebralinear-time-invariantmathematicsphasorphysicsproofhttps://deep0thinking.com/notes/why-physics-equations-are-at-most-second-order/https://deep0thinking.com/notes/why-physics-equations-are-at-most-second-order/How first-derivative actions lead to second-order motion, and what Ostrogradsky’s theorem does—and does not—exclude.Sat, 05 Sep 2026 00:00:00 GMTclassical-mechanicsdifferential-equationlagrangian-mechanicsmathematicsostrogradsky-instabilityphysicshttps://deep0thinking.com/notes/why-the-lorentz-transformation-is-linear/https://deep0thinking.com/notes/why-the-lorentz-transformation-is-linear/A derivation of Lorentz linearity from the structure of inertial frames and spacetime transformations.Sat, 05 Sep 2026 00:00:00 GMTdifferential-geometrylinear-algebramanifoldmathematicsphysicsspecial-relativityhttps://deep0thinking.com/notes/why-use-l2-norm-instead-of-l1-norm-in-loss-functions/https://deep0thinking.com/notes/why-use-l2-norm-instead-of-l1-norm-in-loss-functions/A probabilistic and optimization-based explanation of why squared error is so common in loss functions.Sat, 05 Sep 2026 00:00:00 GMTalgebracomputer-sciencedeep-learninglatexlikelihoodmachine-learningmathematicsoptimizationprobabilityprobability-theoryregression-analysisstatistics