The following individuals have made significant contributions to mathematics without an advanced degree in mathematics. Most had occupations in non-mathematical fields, especially in the early parts of their lives.

• Jean-Robert Argand (1768–1822), a bookstore manager, developed the Argand diagram for representing complex numbers and developed the first rigorous and complete proof of the fundamental theorem of algebra.
• Leon Bankoff (1908–1997), a dentist, was widely respected as an expert in flat geometry, was the editor of the problems department of Pi Mu Epsilon journals, and discovered the Bankoff circle.
• Thomas Bayes (1701–1761), a Presbyterian minister, proved several fundamental theorems in conditional probability and contributed to the theory of asymptotic series.
• Friedrich Bessel (1784–1846), while an accountant’s apprentice, successfully calculated the orbit of Halley’s comet, was later awarded an honorary doctorate on the recommendation of Gauss, and developed Bessel functions while investigating the three-body problem of gravitational theory.
• George Boole (1815–1864), a school teacher with only a primary school education, developed the system of symbolic logic known as Boolean algebra, and contributed to analysis and probability.
• Gerolamo Cardano (1501–1576), a medical doctor, published a paper giving a general solution of cubic and quartic equations, which pioneered the systematic use of negative and imaginary numbers, and he wrote the first systematic work of probability.
• Pierre de Fermat (1601–1665), a lawyer, made pioneering contributions to analytic geometry, calculus, number theory, and probability. He is best known for formulating Fermat’s Last Theorem, which remained unproved until 1994.
• Benjamin Franklin (1706–1790), a printer, statesman, ambassador, inventor, and scientist with almost no formal schooling, developed the Franklin magic square.
• Hermann Grassmann (1809–1877), a secondary mathematics teacher with no university training in mathematics, developed the theories of linear algebra, vector spaces, and Grassman algebras.
• George Green (1793–1841), a miller in a remote part of England, established the mathematical theory of electromagnetism, which included Green’s theorem.
• Oliver Heaviside (1850–1925), a self-educated telegraph operator and electrician, developed transmission line theory, the coaxial cable, and operational calculus, and reformulated Maxwell’s equations of electromagnetism.
• Marin Mersenne (1588–1648), a monk, investigated Mersenne primes, made several mathematical contributions to music theory, and promoted increased contact among scientists.
• Blaise Pascal (1623–1662), educated and supported by his father and his father’s inheritance, developed Pascal’s triangle to present binomial coefficients, made significant contributions to projective geometry, probability theory, developed one of the earliest mechanical calculators, and contributed to physics, philosophy, theology, and literature.
• William Playfair (1759–1823), with a degree in divinity and an eclectic career that included positions as a miller, engineer, silversmith, publicist, and banker, developed the bar chart and pie chart and otherwise showed how charts could communicate numerical data better than tables, and published the first edition of Euclid’s Elements to use algebraic notation.
• Śrīnivāsa Rāmānujan (1887–1920), a clerk who was largely self-educated in mathematics, made highly original contributions to number theory, infinite series, continued fractions, and analysis, was famously mentored by G.H. Hardy, and died at age 32. Like Fermat, he stated many results which were later proved correct.
• Thoralf Skolem (1887–1963) made significant contributions to set theory, mathematical logic, group theory, lattice theory, and Diophantine equations, before obtaining his doctorate in mathematics at age 39.
• Simon Stevin (1548–1620), a clerk, developed a system of decimal fractions which later developed into decimal point notation, and he was instrumental in developing the general notion of a real number.
• François Viète (1540–1603), a lawyer, systematized the notation and presentation of algebra, and developed the first known infinite expression for 𝛑.
• Magnus Wenninger (1919–) a Benedictine monk, built highly accurate models of all 75 uniform polyhedra, many of them for the first time, and has written extensively about polyhedra.
• Caspar Wessel (1745–1818), a surveyor, developed a geometric method of representing complex numbers which preceded Argand’s and was more intuitive but was not widely recognized for almost a century.