A short changelog: when new materials on the foundations of mathematics, logic, probability theory, and algebra appeared on the site.
"The Spectrum of Set Theories" now covers Nelson's Internal Set Theory (IST): a new column in the comparison table, the schemas of transfer, idealization and standardization, and a separate section on two restrictions of comprehension by the form of the formula — stratifiable formulas in NF and internal formulas in IST, together with a concluding section on the impossibility of full homogeneity in set theory (RU).
Published the errata (PDF) for the book "Mathematics as a Foreign Language" — corrections to the Springer edition.
Added "The Spectrum of Set Theories" — a comparison table of ZFC, NBG, ZFA (Aczel's anti-foundation), Quine's NF and two finite models: the status of the universe, foundation, Russell's paradox, Cantor's theorem, predicativity, ordinals, the axiom of choice — and the philosophical positions of Zermelo, Hilbert, von Neumann, Quine, Aczel and Feferman (RU).
Added "A Zoo of Maps Between Models" — partial isomorphism, embedding, elementary embedding, isomorphism, elementary submodel, automorphism, homogeneity, and the Ehrenfeucht–Fraïssé game, all built from a handful of atomic conditions, and the three theorems that connect them non-trivially (RU).
The book "Mathematics as a Foreign Language" is now published by Springer — hardcover and eBook available now (Springer, Barnes & Noble).
Added "The Kissing Number: How Many Apples Fit Around One" — the 1694 Newton-Gregory dispute, an interactive apple-packing widget in the plane, a complete elementary proof that the planar answer is six, and why the space case (twelve) took 259 years to prove (RU).
Added "From Module to Operator" — a module over a ring and its special cases (vector space, algebra), Hamel basis and the theorem on the equipotence of bases (including infinite ones, a consequence of the axiom of choice), coordinates and the isomorphism with Rⁿ, a linear operator and its matrix, the ring of matrices and the dual space; motions and similarities of the plane as a special case of linear operators on R² (RU).
Added "Conformality and Fractional-Linear Transformations" — regularity as local similarity, a derivation of conformality, a proof that inversion sends circles to circles, and Mobius transformations as its composition with similarities. Third and final material in the series, from constructing ℂ to conformal mappings (RU).
Added "Regular Functions" — the complex derivative and Cauchy-Riemann equations, Taylor representability and the uniqueness theorem, Cauchy's integral theorem via Green's formula, the complex exponential and a derivation of Euler's formula. Second material in the series, from constructing ℂ to conformal mappings (RU).
Added "Complex Numbers: Three Constructions of One Field" — R² with coordinatewise arithmetic, the 2×2 real-matrix representation, and R[x]/(x²+1); multiplication by a complex number as rotational homothety. First material in a new series, from constructing ℂ to conformal mappings (RU).
Added "Similarities of the Plane: Homothety and Rotational Homothety" — a continuation of "Motions of the Plane": homothety, rotational homothety, the general fixed-point formula, and the classification of plane similarities (RU).
Added "Motions of the Plane: Chasles' Theorem" — three kinds of plane motion, the three-nails theorem, and a complex-number composition calculator (RU).
Added the English version of "Конструкция Хенкина — интерактивная демонстрация": The Henkin Construction — An Interactive Demonstration.
Added English versions of four materials: The Galton–Watson Process, On Mathematical Induction, Levels of Formalization of Mathematical Analysis, How Mathematical Structures Are Constructed.
Published "One Object, Many Languages" — a periodic table of translations between mathematical theories (RU).
Published the book "Mathematics as a Foreign Language" — chapters A1–C1, in Russian and English (RU).
Added "On Mathematical InductionRU" (English version followed on Jul 29).
Added "The Henkin Construction — An Interactive DemonstrationRU" (English version followed on Jul 29).
Added "The Galton–Watson ProcessRU" — a new "Probability Theory & Statistics" category (English version followed on Jul 29).
Added "How Mathematical Structures Are ConstructedRU" (English version followed on Jul 29).
Added the English version of "Алгебраические структуры · Теория решёток": Algebraic Structures · Lattice Theory.
Added the English version of "Иерархия теорий арифметики второго порядка": Hierarchy of Second-Order Arithmetic Theories.
Added "Hierarchy of Second-Order Arithmetic TheoriesRU" (English version followed on May 26).
Launched the "Materials" section: "Algebraic Structures · Lattice TheoryRU", "Computing the Fixed Point of an Arithmetical FormulaRU", "Levels of Formalization of Mathematical AnalysisRU", "Set Theory: A QuizRU".