Formal & Natural Sciences
Mathematics
The study of structure, quantity, change and space — and the only field where a result, once proved, stays proved.
10 fields · 110 topics · 12 curated works
Fields within Mathematics
Foundations & Logic
12What mathematics rests on, and the limits of what it can prove about itself.
Propositional & Predicate Logic · Set Theory · Model Theory · Proof Theory · Computability & Recursion Theory
Algebra
13The study of operations and the structures they generate, from groups to categories of modules.
Elementary & Abstract Algebra · Group Theory · Ring Theory · Field Theory · Galois Theory
Number Theory
12The behaviour of the integers — the oldest questions in mathematics and several of the hardest.
Elementary Number Theory · Analytic Number Theory · Algebraic Number Theory · Diophantine Equations · Distribution of Primes
Analysis
11Limits, continuity and infinite processes made rigorous.
Real Analysis · Complex Analysis · Functional Analysis · Measure Theory · Harmonic & Fourier Analysis
Geometry
10Space, shape and the invariants that survive deformation.
Euclidean Geometry · Non-Euclidean Geometry · Differential Geometry · Riemannian Geometry · Algebraic Geometry
Topology
10The properties of space that survive stretching but not tearing.
Point-Set Topology · Algebraic Topology · Differential Topology · Homotopy Theory · Homology & Cohomology
Combinatorics & Discrete Mathematics
10Counting, arrangement and the structure of finite systems.
Enumerative Combinatorics · Graph Theory · Extremal Combinatorics · Ramsey Theory · Design Theory
Probability Theory
11The mathematics of uncertainty and random structure.
Measure-Theoretic Probability · Stochastic Processes · Markov Chains · Martingales · Random Walks
Applied Mathematics
12Mathematics aimed at systems that actually exist.
Numerical Analysis · Optimisation Theory · Dynamical Systems · Chaos Theory · Control Theory
History & Philosophy of Mathematics
9Where the ideas came from, and what mathematical truth even is.
Ancient & Greek Mathematics · Islamic Golden Age Mathematics · The Calculus Priority Dispute · The Foundational Crisis · Platonism vs Formalism
Reading in Mathematics
12Start here
No prior grounding assumed.
- BookA Mathematician's ApologyG. H. Hardy, 1940
A working mathematician's defence of pure mathematics as an art form judged by beauty rather than utility.
- BookHow to Solve ItGeorge Pólya, 1945
A general heuristic for attacking any problem: understand it, plan, execute, then look back at what the solution taught you.
Then
Assumes you know the vocabulary.
- BookElementsEuclid, 300 BCE
Derives the whole of classical geometry from five postulates, and in doing so invents the axiomatic method every later mathematics is written in.
- EssayThe Unreasonable Effectiveness of Mathematics in the Natural SciencesEugene Wigner, 1960· 14 pages
Asks why mathematics invented for its own sake keeps turning out to describe physical reality, and offers no comfortable answer.
- BookMathematical Methods in the Physical SciencesMary L. Boas, 1966
Collects every mathematical technique a physical scientist actually uses into one book, chosen for utility rather than for elegance.
Go deeper
Primary sources and full treatments.
- PaperOn Computable Numbers, with an Application to the EntscheidungsproblemAlan Turing, 1936· 36 pages
Defines the universal machine and proves the halting problem undecidable — the founding document of computer science.
- BookPrinciples of Mathematical AnalysisWalter Rudin, 1953
The standard first course in rigorous analysis, austere to the point of severity — it teaches proof by refusing to do any of the work for you.
- BookConcrete MathematicsGraham, Knuth & Patashnik, 1989
The discrete mathematics needed to actually analyse algorithms — sums, recurrences and generating functions, taught as a manipulable craft.
- BookAn Introduction to Probability Theory and Its ApplicationsWilliam Feller, 1950
The classic treatment, built on problems rather than measure theory, and still the best source of probabilistic intuition.
- BookTopologyJames Munkres, 1975
The standard introduction to point-set topology, patient enough that the definitions feel motivated rather than imposed.
- BookAlgebraMichael Artin, 1991
Builds abstract algebra out of linear algebra and matrix groups, so the axioms arrive as descriptions of things you already understand.
- BookProofs from THE BOOKMartin Aigner & Günter Ziegler, 1998
A collection of the most elegant known proofs of significant theorems, chosen for economy rather than generality.