Glossary

54 terms used across this atlas, defined precisely. Each entry states what the term means and then the condition people leave out, because a definition that is almost right is the kind that costs you an afternoon.

Foundations and logic

• Axiom — A statement assumed without proof, taken as a starting point from which other results are derive…
• Conjecture — A precisely stated mathematical claim that is expected to be true but has neither been proved no…
• Counterexample — A single case satisfying a claim’s hypotheses but failing its conclusion, which is enough to dis…
• Formal system — A precisely specified language together with axioms and inference rules, in which proofs are fin…
• Gödel’s incompleteness theorems — Two results of Kurt Gödel (1931). First: any consistent, effectively axiomatised formal system s…
• Independence — A statement is independent of an axiom system when neither it nor its negation can be proved fro…
• Open problem — A question that has not been settled: no accepted proof, no disproof, and for questions that are…
• Proof — A finite chain of deductive steps, each justified by an axiom or an already established result, …
• Theorem — A statement established by a valid deductive proof from stated axioms together with previously p…
• Undecidability — In computability, the property of a decision problem for which no algorithm always halts with a …

Set theory

• Axiom of choice — The assumption that for any collection of nonempty sets there is a function selecting exactly on…
• Cardinality — A measure of the size of a set. Two sets have the same cardinality when their elements can be ma…
• Continuum hypothesis — The statement that no set has cardinality strictly between that of the natural numbers and that …
• Countable set — A set whose elements can be listed in a sequence indexed by the natural numbers, possibly termin…
• Set theory — The study of sets as a foundation for mathematics, including infinite cardinal and ordinal numbe…

Number theory

• abc conjecture — For coprime positive integers with a + b = c, the conjecture that for every ε > 0 only finitely …
• Amicable numbers — A pair of distinct positive integers, each equal to the sum of the other’s proper divisors. The …
• Collatz conjecture — The conjecture that for every positive integer n, repeatedly applying the rule "halve n if it is…
• Diophantine equation — A polynomial equation, or system of equations, with integer coefficients, for which integer solu…
• Elliptic curve — A smooth projective curve of genus 1 together with a chosen rational point on it. Concretely, th…
• Goldbach conjecture — The conjecture that every even integer greater than 2 is a sum of two primes. The two primes nee…
• Mersenne prime — A prime of the form 2ⁿ − 1 for some integer n…
• Modular arithmetic — Arithmetic in which numbers wrap around on reaching a fixed modulus n, so that a and b are treat…
• Perfect number — A positive integer equal to the sum of its proper divisors, meaning its positive divisors strict…
• Prime number — A positive integer greater than 1 whose only positive divisors are 1 and itself, equivalently an…
• Prime number theorem — The theorem that the number of primes not exceeding x is asymptotic to x / ln x, meaning the rat…
• Riemann hypothesis — The conjecture that every nontrivial zero of the Riemann zeta function has real part exactly 1/2…
• Riemann zeta function — The function ζ(s) = ∑ 1/nˢ summed over positive integers n, defined by that series where it conv…
• Twin prime — A prime that differs by 2 from another prime. The two together, such as 11 and 13, are called a …

Analysis

• Analytic continuation — The extension of a complex function beyond the region where its original formula converges, to a…
• Critical line — The vertical line in the complex plane consisting of numbers with real part 1/2, on which the Ri…
• Transcendental number — A real or complex number that is not a root of any nonzero polynomial with rational coefficients…

Geometry and topology

• Euler characteristic — A number attached to a space, computed for a polyhedron as vertices minus edges plus faces, and …
• Genus — For a closed orientable surface, the number of holes: a sphere has genus 0, a torus genus 1, a t…
• Homeomorphism — A continuous one-to-one correspondence between two spaces whose inverse is also continuous. Spac…
• Knot theory — The study of closed loops embedded in three-dimensional space, considered equivalent when one ca…
• Manifold — A space in which every point has a neighbourhood resembling ordinary n-dimensional Euclidean spa…
• Poincaré conjecture — The statement, now a theorem, that every closed simply connected three-dimensional manifold is h…
• Topology — The study of properties of a space preserved under continuous deformation — stretching and bendi…

Graph theory and combinatorics

• Chromatic number — The smallest number of colours needed to colour a graph’s vertices so that no two adjacent verti…
• Graph — A set of vertices together with a set of edges, each edge joining a pair of vertices…
• Hamiltonian path — A path through a graph visiting every vertex exactly once. A Hamiltonian cycle additionally retu…
• Pigeonhole principle — If more items are placed into containers than there are containers, then some container holds at…
• Planar graph — A graph that can be drawn in the plane with no two edges crossing…
• Ramsey theory — The study of the order that must appear in any sufficiently large structure, however that struct…

Theoretical computer science

• Algorithm — A finite, precisely specified procedure that transforms an input into an output in a finite numb…
• Decision problem — A computational problem whose answer for each input is yes or no…
• NP-complete — A problem in NP to which every other NP problem can be reduced in polynomial time, making it at …
• P versus NP — The question of whether every decision problem whose yes-instances can be verified in polynomial…
• Polynomial time — A running time bounded by some fixed power of the input size. Problems solvable this way form th…

Dynamical systems

• Attractor — A set of states that is invariant under the dynamics, attracts all nearby starting states over t…
• Dynamical system — A set of possible states together with a fixed rule determining how a state evolves over time…
• Ergodic theory — The study of the long-run statistical behaviour of measure-preserving dynamical systems, where a…
• Orbit — The sequence or path of states produced by repeatedly applying a dynamical system’s rule to a gi…