Formal & Natural Sciences
The study of structure, quantity, change and space — and the only field where a result, once proved, stays proved.
10 fields · 120 topics · 178 curated works
What mathematics rests on, and the limits of what it can prove about itself.
Foundations & Overviews · Propositional & Predicate Logic · Set Theory · Model Theory · Proof Theory
The study of operations and the structures they generate, from groups to categories of modules.
Foundations & Overviews · Elementary & Abstract Algebra · Group Theory · Ring Theory · Field Theory
The behaviour of the integers — the oldest questions in mathematics and several of the hardest.
Foundations & Overviews · Elementary Number Theory · Analytic Number Theory · Algebraic Number Theory · Diophantine Equations
Limits, continuity and infinite processes made rigorous.
Foundations & Overviews · Real Analysis · Complex Analysis · Functional Analysis · Measure Theory
Space, shape and the invariants that survive deformation.
Foundations & Overviews · Euclidean Geometry · Non-Euclidean Geometry · Differential Geometry · Riemannian Geometry
The properties of space that survive stretching but not tearing.
Foundations & Overviews · Point-Set Topology · Algebraic Topology · Differential Topology · Homotopy Theory
Counting, arrangement and the structure of finite systems.
Foundations & Overviews · Enumerative Combinatorics · Graph Theory · Extremal Combinatorics · Ramsey Theory
The mathematics of uncertainty and random structure.
Foundations & Overviews · Measure-Theoretic Probability · Stochastic Processes · Markov Chains · Martingales
Mathematics aimed at systems that actually exist.
Foundations & Overviews · Numerical Analysis · Optimisation Theory · Dynamical Systems · Chaos Theory
Where the ideas came from, and what mathematical truth even is.
Foundations & Overviews · Ancient & Greek Mathematics · Islamic Golden Age Mathematics · The Calculus Priority Dispute · The Foundational Crisis
No prior grounding assumed.
The Thirteen Books of Euclid's Elements
Euclid (trans. Thomas L. Heath) · 300 BCE
Builds the whole of classical geometry and much of number theory from five postulates and a handful of common notions, establishing the axiomatic…
+29 more at this level
Assumes you know the vocabulary.
Elements
Euclid · 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.
+46 more at this level
Primary sources and full treatments.
On Computable Numbers, with an Application to the Entscheidungsproblem
Alan Turing · 1936
Defines the universal machine and proves the halting problem undecidable — the founding document of computer science.
+100 more at this level
12 of 178 works
Euclid (trans. Thomas L. Heath)
Builds the whole of classical geometry and much of number theory from five postulates and a handful of common notions, establishing the axiomatic method as the model for what a mathematical proof is.
link checked 17 Sept 2026Leonhard Euler
Solves the Konigsberg bridges puzzle by discarding distance and shape entirely and reasoning only about which crossings connect which landmasses, the founding argument of graph theory.
link checked 17 Sept 2026David Brewster
Draws on Newton's papers and correspondence to argue that his scientific achievement and his private, often bitter disputes over priority came from the same intensely secretive and self-certain temperament.
about 7 hours to readlink checked 17 Sept 2026Edwin A. Abbott
A two-dimensional narrator's encounter with a sphere argues, through satire, that a being confined to one dimension cannot conceive of the next by reason alone.
about 2.5 hours to readlink checked 17 Sept 2026W.W. Rouse Ball
Traces mathematics from ancient Egypt through the calculus era as a continuous chain of individuals building on and correcting each other, arguing the history is best told through its practitioners rather than as an abstract progression of ideas.
link checked 17 Sept 2026Henri Poincare
Argues that mathematical discovery is driven by an unconscious aesthetic sense for which combinations of ideas are fruitful, against the view that proof is generated by mechanical logical deduction alone.
link checked 17 Sept 2026Derrick Norman Lehmer
Builds the projective plane from incidence and cross-ratio alone, with no coordinates, arguing that Desargues's and Pappus's theorems and the principle of duality are more fundamental than any metric structure laid on top of them.
link checked 17 Sept 2026Notes on this copyG.H. Hardy
Defends pure mathematics done for its own beauty rather than its use, arguing a mathematician's work is judged by seriousness and depth in the same way a poem or painting is, not by application.
56 pageslink checked 17 Sept 2026Ernest Nagel & James R. Newman
Walks a general reader through Gödel's argument that any consistent system strong enough for arithmetic contains truths it cannot prove.
link checked 17 Sept 2026Joseph L. Doob
Explains, for a general mathematical audience, why the martingale property that tomorrow's expected value is today's value is the single idea unifying gambling systems, random walks and the convergence theorems built on them.
link checked 17 Sept 2026Peter G. Doyle & J. Laurie Snell
Shows that the trajectory of a random walk on a graph and the flow of current through the same graph as a resistor network are the same mathematics seen twice, using each to build intuition for the other.
link checked 17 Sept 2026Sidney A. Morris
Builds point-set topology from metric spaces toward the general definition of a topological space, arguing each axiom should be motivated by a concrete example before it is stated abstractly.
link checked 17 Sept 2026