Human Tech Tree

Formal Sciences & Matter

Mathematics

63 points from the earliest roots to the research frontier: 46 researched, 3 current research, 7 unsolved and 7 that become possible once they are solved. State of knowledge: October 2026.

Open the interactive tree →

Prehistorybefore 3000 BC

  • Tally Marks & Counting~43,000 years agoResearchedNotches on bones record numbers: the oldest known step from counting in the head to counting in objects.

Antiquity3000 BC – 500 AD

  • Number Systems & Place Value~3000-2000 BCResearchedBabylonian scribes compute in a base-60 place-value system, the foundation of bookkeeping, astronomy and all later mathematics.
  • Deductive Proof (Greece)~600 BCResearchedGreek thinkers from Thales to Pythagoras demand that mathematical claims be proven from earlier ones, not just observed.
  • Geometry (Euclid)~300 BCResearchedEuclid's Elements build geometry from a few axioms and make mathematical proof the standard method of the discipline.
  • Trigonometry~150 BCResearchedHipparchus' table of chords links angles to lengths; Ptolemy, Indian and Arab mathematicians turn it into sines and cosines.
  • Nine Chapters (China)~100 ADResearchedThe Chinese Nine Chapters on the Mathematical Art solve systems of linear equations with a method equal to Gaussian elimination.

Middle Ages & Renaissance500 – 1700

  • Zero, Decimals & Algebra~820ResearchedIndian and Arab scholars establish zero as a number, the decimal system and algebra, the art of computing with unknowns.
  • Complex Numbers1545ResearchedCardano's solution of the cubic equation (1545) forces mathematicians to compute with roots of negative numbers.
  • Logarithms (Napier)1614ResearchedNapier's tables (1614) turn multiplication into addition and halve the labour of astronomy and navigation.
  • Analytic Geometry (Descartes)1637ResearchedCoordinates turn curves into equations and equations into curves: algebra and geometry become one language.
  • Probability Theory1654ResearchedPascal and Fermat found the mathematics of chance in a correspondence about a dice game, the basis of statistics, insurance and AI.
  • Calculus (Analysis)1684ResearchedNewton and Leibniz independently develop derivatives and integrals, the tool for describing change exactly.

Industrial Age1700 – 1900

  • Graph Theory (Euler)1736ResearchedEuler's solution of the Königsberg bridge problem (1736) founds the study of networks of points and links.
  • Partial Differential Equations1747ResearchedD'Alembert's wave equation (1747) begins the use of differential equations to describe continuous fields and media.
  • Bayesian Inference1763ResearchedBayes' theorem (published 1763) shows how to update a belief when new evidence arrives.
  • Number Theory & Group Theory1801ResearchedGauss founds modern number theory (1801) and Galois group theory (1832): the mathematics of primes and of symmetry.
  • Statistics & Least Squares1809ResearchedLeast squares, the normal curve and population statistics (1805-1835) let scientists extract reliable signals from noisy data.
  • Fourier Analysis1822ResearchedFourier shows (1822) that any signal can be built from sine waves, the working language of heat, sound, light and data.
  • Non-Euclidean Geometry1829ResearchedLobachevsky, Bolyai and Riemann show that geometry also works on curved spaces, later the language of relativity.
  • Gradient Descent1847ResearchedCauchy's method of steepest descent (1847) finds a minimum by repeatedly stepping downhill.
  • Boolean Algebra1854ResearchedGeorge Boole shows that logic can be calculated like algebra with just two values, true and false (1854).
  • Riemannian Geometry1854ResearchedRiemann's 1854 lecture defines curved spaces of any dimension by a local measure of distance.
  • Linear Algebra & Matrices1858ResearchedCayley's matrix algebra (1858) gives a compact calculus for systems of equations, rotations and transformations.
  • Set Theory & Formal Logic1874ResearchedCantor creates set theory (1874) and Frege formal logic (1879): the attempt to put all of mathematics on a secure foundation.
  • Topology1895ResearchedPoincaré's Analysis Situs (1895) studies properties of shapes that survive stretching and bending, such as holes and connectedness.
  • Prime Number Theorem1896ResearchedHadamard and de la Vallée Poussin prove (1896) that primes thin out like 1/ln n, using Riemann's complex zeta function.

Machine Age1900 – 1945

  • Hilbert's Problems & Program1900ResearchedIn 1900 Hilbert lists 23 open problems and later asks for a complete, provably consistent foundation of mathematics.
  • Measure-Theoretic Probability1902-1933ResearchedLebesgue's measure and integral (1902) and Kolmogorov's axioms (1933) put probability on a rigorous footing, the base of modern statistics.
  • Markov Chains1906ResearchedMarkov's chains (1906) model sequences in which the next state depends only on the present one.
  • Abstract Algebra (Noether)1921-1931ResearchedEmmy Noether's 1921 work on rings and van der Waerden's Moderne Algebra (1930-31) recast algebra as the study of structures: groups, rings, fields…
  • Gödel's Incompleteness1931ResearchedGödel proves in 1931 that every sufficiently strong system of rules contains true statements it cannot prove.
  • Computability (Turing)1936ResearchedTuring describes the universal computing machine in 1936 and proves that some problems cannot be solved by any algorithm.

Atomic & Space Age1945 – 1990

  • Linear Optimization (Simplex)1947ResearchedDantzig invents the simplex method in 1947, the standard way to get the best result from scarce resources in logistics, flights and production.
  • Monte Carlo Methods1949ResearchedUlam, von Neumann and Metropolis use random sampling on a computer to solve problems too hard for formulas (1946-49).
  • Chaos Theory1963ResearchedLorenz discovers by computer in 1963 that tiny differences in starting values change a system completely, which limits weather forecasts.
  • Fast Fourier Transform1965ResearchedCooley and Tukey's algorithm (1965) computes Fourier transforms thousands of times faster and makes digital signal processing practical.
  • Complexity Theory & NP1971ResearchedCook and Karp show in 1971/72 that many hard search problems are equally hard; the question 'P versus NP' is born.
  • Four Colour Theorem Proved1976ResearchedAppel and Haken prove (1976) that four colours suffice for any map, the first major theorem proved with a computer.

Digital Age1990 – 2015

  • Fermat's Last Theorem Proved1995ResearchedAndrew Wiles proves Fermat's Last Theorem in 1994/95, after 358 years, with tools Fermat never knew.
  • Kepler Conjecture & Flyspeck1998-2014ResearchedThomas Hales's computer-assisted proof (1998) that stacked spheres fill at most 74% of space was checked line by line by the Flyspeck project (2014).
  • Poincaré Conjecture Proved2003ResearchedPerelman proves (2002-03) the Poincaré conjecture with Hamilton's Ricci flow, the first Millennium Problem to be solved.

Present2015 – Oct 2026

  • Formal Proofs (Lean, Rocq)2005-2026Current researchProof assistants check mathematics without gaps; in September 2026 Anthropic reported that Claude agents formalized Fermat's Last Theorem in Lean in…
  • Sphere Packing in 8 & 24 Dims2016ResearchedMaryna Viazovska proved in 2016 that the E8 and Leech lattices pack equal spheres as densely as possible in 8 and 24 dimensions; Fields Medal 2022.
  • Busy Beaver BB(5) Proved2024ResearchedAn online collective proves in 2024 the fifth Busy Beaver value (47,176,870 steps), fully machine-checked in Rocq.
  • Fluid Equations from Atoms2024-2025ResearchedDeng, Hani and Ma derived Boltzmann's gas equation from hard-sphere dynamics for arbitrarily long times, a main step in Hilbert's sixth problem.
  • Geometric Langlands Proved2024ResearchedGaitsgory, Raskin and collaborators prove the geometric Langlands conjecture in 2024, bridging number theory, geometry and physics.
  • AI Wins Math Olympiad Gold2025Current researchIn July 2025 AI systems reach gold level at the International Mathematical Olympiad (35/42); in July 2026 Huawei and Xiaohongshu claim perfect 42/42…
  • Kakeya Conjecture in 3D2025ResearchedHong Wang and Joshua Zahl prove the three-dimensional Kakeya conjecture in February 2025; Wang receives a Fields Medal in 2026.
  • AI Proves Open Problems2026Current researchSince January 2026 AI systems are credited with settling open questions: Erdős problems, the unit-distance conjecture (May), a Riemann-zero bound…

Research Frontier · Todayunsolved as of Oct 2026

  • All Mathematics Machine-CheckedopenUnsolvedHaving every important theorem checked gap-free by machine is still far off; in September 2026 AI agents formalized the whole proof of Fermat's Last…
  • Birch and Swinnerton-DyeropenUnsolvedDoes the number of rational points on an elliptic curve match the behaviour of its L-function at 1? The rank part is proved only when that behaviour…
  • Hodge ConjectureopenUnsolvedIs every Hodge class on a smooth projective variety a combination of algebraic cycles? Proved only in special cases, such as abelian fourfolds in…
  • Navier-Stokes ProblemopenUnsolvedDo flows of water and air stay smooth forever, or can they blow up in finite time? In September 2026 OpenAI claimed a blow-up proof, but only with…
  • P versus NPopenUnsolvedIf a solution can be checked quickly, can it also be found quickly? Open since 1971; a proof earns one million dollars.
  • Riemann HypothesisopenUnsolvedDo all non-trivial zeros of the zeta function lie on one line? Open since 1859; it governs how primes are distributed.
  • Yang-Mills Mass GapopenUnsolvedProve that quantum Yang-Mills theory exists in four dimensions and has a mass gap; simulations support it, but no mathematical proof exists.

If Solvedbecomes possible

  • AI Research Mathematician2030s?If solvedAI systems develop new theories on their own, with machine-checked proofs, and change how mathematics is done.
  • Algebraic Cycles UnderstoodopenIf solvedA proof would link the topology of algebraic varieties to the shapes that polynomial equations can cut out.
  • Flow Simulation With GuaranteesopenIf solvedOnce it is clear when Navier-Stokes solutions stay smooth, flow programs can come with mathematical error guarantees.
  • Primes Fully UnderstoodopenIf solvedWith the Riemann hypothesis proven, it is clear how regularly primes are scattered, and many conditional theorems become unconditional.
  • Provably Secure EncryptionopenIf solvedEncryption whose security rests on a mathematical proof instead of unproven assumptions.
  • Rational Solutions PredictableopenIf solvedFor any elliptic curve one can compute from its L-function whether and how many rational solutions exist, with proof.
  • Rigorous Quantum Field TheoryopenIf solvedQuantum Yang-Mills theory in four dimensions is built and checked mathematically, with a proven mass gap.