Biographies of the greatest mathematicians are in separate files by birth year:
|Born before 400||Born betw. 400 & 1559||Born betw. 1560 & 1699|
|Born betw. 1700 & 1799||Born betw. 1800 & 1859 (this page)||Born betw. 1860 & 1975|
|List of Greatest Mathematicians|
Julius Plücker (1801-1868) Germany
Plücker was one of the most innovative geometers, inventing line geometry (extending the atoms of geometry beyond just points), enumerative geometry (which considered such questions as the number of loops in an algebraic curve), geometries of more than three dimensions, and generalizations of projective geometry. He also gave an improved theoretic basis for the Principle of Duality. His novel methods and notations were important to the development of modern analytic geometry, and inspired Cayley, Klein and Lie. He resolved the famous Cramer-Euler Paradox and the related Poncelet Paradox by studying the singularities of curves; Cayley described this work as "most important ... beyond all comparison in the entire subject of modern geometry." In part due to conflict with his more famous rival, Jakob Steiner, Plücker was under-appreciated in his native Germany, but achieved fame in France and England. In addition to his mathematical work in algebraic and analytic geometry, Plücker did significant work in physics, e.g. his work with cathode rays. Although less brilliant as a theorem prover than Steiner, Plücker's work, taking full advantage of analysis and seeking physical applications, was far more influential.
Niels Henrik Abel (1802-1829) Norway
At an early age, Niels Abel studied the works of the greatest mathematicians, found flaws in their proofs, and resolved to reprove some of these theorems rigorously. He was the first to fully prove the general case of Newton's Binomial Theorem, one of the most widely applied theorems in mathematics. Several important theorems of analysis are named after Abel, including the (deceptively simple) Abel's Theorem of Convergence (published posthumously). Along with Galois, Abel is considered one of the two founders of group theory. Abel also made contributions in algebraic geometry and the theory of equations.
Inversion (replacing y = f(x) with x = f-1(y)) is a key idea in mathematics (consider Newton's Fundamental Theorem of Calculus); Abel developed this insight. Legendre had spent much of his life studying elliptic integrals, but Abel inverted these to get elliptic functions, and was first to observe (but in a manuscript mislaid by Cauchy) that they were doubly periodic. Elliptic functions quickly became a productive field of mathematics, and led to more general complex-variable functions, which were important to the development of both abstract and applied mathematics.
Finding the roots of polynomials is a key mathematical problem: the general solution of the quadratic equation was known by ancients; the discovery of general methods for solving polynomials of degree three and four is usually treated as the major math achievement of the 16th century; so for over two centuries an algebraic solution for the general 5th-degree polynomial (quintic) was a Holy Grail sought by most of the greatest mathematicians. Abel proved that most quintics did not have such solutions. This discovery, at the age of only nineteen, would have quickly awed the world, but Abel was impoverished, had few contacts, and spoke no German. When Gauss received Abel's manuscript he discarded it unread, assuming the unfamiliar author was just another crackpot trying to square the circle or some such. His genius was too great for him to be ignored long, but, still impoverished, Abel died of tuberculosis at the age of twenty-six. His fame lives on and even the lower-case word 'abelian' is applied to several concepts. Liouville said Abel was the greatest genius he ever met. Hermite said "Abel has left mathematicians enough to keep them busy for 500 years."
Carl G. J. Jacobi (1804-1851) Germany
Jacobi was a prolific mathematician who did decisive work in the algebra and analysis of complex variables, and did work in number theory (e.g. cubic reciprocity) which excited Carl Gauss. He is sometimes described as the successor to Gauss. As an algorist (manipulator of involved algebraic expressions), he may have been surpassed only by Euler and Ramanujan. He was also a very highly regarded teacher. In mathematical physics, Jacobi perfected Hamilton's principle of stationary action, and made other important advances.
Jacobi's most significant early achievement was the theory of elliptic functions, e.g. his fundamental result about functions with multiple periods. Jacobi was the first to apply elliptic functions to number theory, extending Lagrange's famous Four-Squares Theorem to show in how many distinct ways a given integer can be expressed as the sum of four squares. He also made important discoveries in many other areas including theta functions (e.g. his Jacobi Triple Product Identity), higher fields, number theory, algebraic geometry, differential equations, q-series, hypergeometric series, determinants, Abelian functions, and dynamics. He devised the algorithms still used to calculate eigenvectors and for other important matrix manipulations. The range of his work is suggested by the fact that the "Hungarian method," an efficient solution to an optimization problem published more than a century after Jacobi's death, has since been found among Jacobi's papers.
Like Abel, as a young man Jacobi attempted to factor the general quintic equation. Unlike Abel, he seems never to have considered proving its impossibility. This fact is sometimes cited to show that despite Jacobi's creativity, his ill-fated contemporary was the more brilliant genius.
Johann Peter Gustav Lejeune Dirichlet (1805-1859) Germany
Dirichlet was preeminent in algebraic and analytic number theory, but did advanced work in several other fields as well: He discovered the modern definition of function, the Voronoi diagram of geometry, and important concepts in differential equations, topology, and statistics. His proofs were noted both for great ingenuity and unprecedented rigor. As an example of his careful rigor, he found a fundamental flaw in Steiner's Isoperimetric Theorem proof which no one else had noticed. In addition to his own discoveries, Dirichlet played a key role in interpreting the work of Gauss, and was an influential teacher, mentoring famous mathematicians like Bernhard Riemann (who considered Dirichlet second only to Gauss among living mathematicians), Leopold Kronecker and Gotthold Eisenstein.
As an impoverished lad Dirichlet spent his money on math textbooks; Gauss' masterwork became his life-long companion. Fermat and Euler had proved the impossibility of xk + yk = zk for k = 4 and k = 3; Dirichlet became famous by proving impossibility for k = 5 at the age of 20. Later he proved the case k = 14 and, later still, may have helped Kummer extend Dirichlet's quadratic fields, leading to proofs of more cases. More important than his work with Fermat's Last Theorem was his Unit Theorem, considered one of the most important theorems of algebraic number theory. The Unit Theorem is unusually difficult to prove; it is said that Dirichlet discovered the proof while listening to music in the Sistine Chapel. A key step in the proof uses Dirichlet's Pigeonhole Principle, a trivial idea but which Dirichlet applied with great ingenuity.
Dirichlet did seminal work in analysis and is considered the founder of analytic number theory. He invented a method of L-series to prove the important theorem (Gauss' conjecture) that any arithmetic series (without a common factor) has an infinity of primes. It was Dirichlet who proved the fundamental Theorem of Fourier series: that periodic analytic functions can always be represented as a simple trigonometric series. Although he never proved it rigorously, he is especially noted for the Dirichlet's Principle which posits the existence of certain solutions in the calculus of variations, and which Riemann found to be particularly fruitful. Other fundamental results Dirichlet contributed to analysis and number theory include a theorem about Diophantine approximations and his Class Number Formula.
William Rowan (Sir) Hamilton (1805-1865) Ireland
Hamilton was a childhood prodigy. Home-schooled and self-taught, he started as a student of languages and literature, was influenced by an arithmetic prodigy his own age, read Euclid, Newton and Lagrange, found an error by Laplace, and made new discoveries in optics; all this before the age of seventeen when he first attended school. At college he enjoyed unprecedented success in all fields, but his undergraduate days were cut short abruptly by his appointment as Royal Astronomer of Ireland at the age of 22. He soon began publishing his revolutionary treatises on optics, in which he developed Hamilton's Principle of Stationary Action. This Principle refined and corrected the earlier principles of least action developed by Maupertuis, Fermat, and Euler; it (and related principles) are key to much of modern physics. His early writing also predicted that some crystals would have an hitherto unknown "conical" refraction mode; this was soon confirmed experimentally.
Hamilton's Principle of Least Action, and its associated equations and concept of configuration space, led to a revolution in mathematical physics. Since Maupertuis had named this Principle a century earlier, it is possible to underestimate Hamilton's contribution. However Maupertuis, along with others credited with anticipating the idea (Fermat, Leibniz, Euler and Lagrange) failed to state the full Principle correctly. Rather than minimizing action, physical systems sometimes achieve a non-minimal but stationary action in configuration space. (Poisson and d' Alembert had noticed exceptions to Euler-Lagrange least action, but failed to find Hamilton's solution. Jacobi also deserves some credit for the Principle, but his work came after reading Hamilton.) Because of this Principle, as well as his wave-particle duality (which would be further developed by Planck and Einstein), Hamilton can be considered a major early influence on quantum theory.
Hamilton also made revolutionary contributions to dynamics, differential equations, the theory of equations, numerical analysis, fluctuating functions, and graph theory (he marketed a puzzle based on his Hamiltonian paths). He invented the ingenious hodograph. He coined several mathematical terms including vector, scalar, associative, and tensor. In addition to his brilliance and creativity, Hamilton was renowned for thoroughness and produced voluminous writings on several subjects.
Hamilton himself considered his greatest accomplishment to be the development of quaternions, a non-Abelian field to handle 3-D rotations. While there is no 3-D analog to the Gaussian complex-number plane (based on the equation i2 = -1 ), quaternions derive from a 4-D analog based on i2 = j2 = k2 = ijk = -jik = -1. Although matrix and tensor methods may seem more general, quaternions are still in wide engineering use because of practical advantages, e.g. avoidance of "gimbal lock."
Hamilton once wrote: "On earth there is nothing great but man; in man there is nothing great but mind."
Hermann Günter Grassmann (1809-1877) Germany
Grassmann was an exceptional polymath: the term Grassmann's Law is applied to two separate facts in the fields of optics and linguistics, both discovered by Hermann Grassmann. He also did advanced work in crystallography, electricity, botany, folklore, and also wrote on political subjects. He had little formal training in mathematics, yet single-handedly developed linear algebra, vector and tensor calculus, multi-dimensional geometry, new results about cubic surfaces, the theory of extension, and exterior algebra; most of this work was so innovative it was not properly appreciated in his own lifetime. (Heaviside rediscovered vector analysis many years later.) Grassmann's exterior algebra, and the associated concept of Grassmannian manifold, provide a simplifying framework for many algebraic calculations. Recently their use led to an important simplification in quantum physics calculations.
Of his linear algebra, one historian wrote "few have come closer than Hermann Grassmann to creating, single-handedly, a new subject." Important mathematicians inspired directly by Grassmann include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.
Joseph Liouville (1809-1882) France
Liouville did expert research in several areas including number theory, differential geometry, complex analysis (especially Sturm-Liouville theory, boundary value problems and dynamical analysis), harmonic functions, topology and mathematical physics. Several theorems bear his name, including the key result that any bounded entire function must be constant (the Fundamental Theorem of Algebra is an easy corollary of this!); important results in differential equations, differential algebra, differential geometry; a key result about conformal mappings; and an invariance law about trajectories in phase space which leads to the Second Law of Thermodynamics and is key to Hamilton's work in physics. He was first to prove the existence of transcendental numbers. (His proof was constructive, unlike that of Cantor which came 30 years later). He invented Liouville integrability and fractional calculus; he found a new proof of the Law of Quadratic Reciprocity. In addition to multiple Liouville Theorems, there are two "Liouville Principles": a fundamental result in differential algebra, and a fruitful theorem in number theory. Liouville was hugely prolific in number theory but this work is largely overlooked, e.g. the following remarkable generalization of Aryabhata's identity:
for all N, Σ (da3) = (Σ da)2
where da is the number of divisors of a, and the sums are taken over all divisors a of N.
Liouville established an important journal; influenced Catalan, Jordan, Chebyshev, Hermite; and helped promote other mathematicians' work, especially that of Évariste Galois, whose important results were almost unknown until Liouville clarified them. In 1851 Augustin Cauchy was bypassed to give a prestigious professorship to Liouville instead.
Ernst Eduard Kummer (1810-1893) Germany
Despite poverty, Kummer became an important mathematician at an early age, doing work with hypergeometric series, functions and equations, and number theory. He worked on the 4-degree Kummer Surface, an important algebraic form which inspired Klein's early work. He solved the ancient problem of finding all rational quadrilaterals. His most important discovery was ideal numbers; this led to the theory of ideals and p-adic numbers; this discovery's revolutionary nature has been compared to that of non-Euclidean geometry. Kummer is famous for his attempts to prove, with the aid of his ideal numbers, Fermat's Last Theorem. He established that theorem for almost all exponents (including all less than 100) but not the general case.
Kummer was an inspirational teacher; his famous students include Cantor, Frobenius, Fuchs, Schwarz, Gordan, Joachimsthal, Bachmann, and Kronecker. (Leopold Kronecker was a brilliant genius sometimes ranked ahead of Kummer in lists like this; that Kummer was Kronecker's teacher at high school persuades me to give Kummer priority.)
Évariste Galois (1811-1832) France
Galois, who died before the age of twenty-one, not only never became a professor, but was barely allowed to study as an undergraduate. His output of papers, mostly published posthumously, is much smaller than most of the others on this list, yet it is considered among the most awesome works in mathematics. He applied group theory to the theory of equations, revolutionizing both fields. (Galois coined the mathematical term group.) While Abel was the first to prove that some polynomial equations had no algebraic solutions, Galois established the necessary and sufficient condition for algebraic solutions to exist. His principal treatise was a letter he wrote the night before his fatal duel, of which Hermann Weyl wrote: "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."
Galois' ideas were very far-reaching; for example he is credited as first to prove that trisecting a general angle with Plato's rules is impossible. Galois is sometimes cited (instead of Archimedes, Gauss or Ramanujan) as "the greatest mathematical genius ever." But he was too far ahead of his time -- the top mathematicians of his day rejected his theory as "incomprehensible." Galois was persecuted for his Republican politics, imprisoned, and forced to fight a duel, where he was left to bleed out without medical attention. His last words (spoken to his brother) were "Ne pleure pas, Alfred! J'ai besoin de tout mon courage pour mourir à vingt ans!" This tormented life, with its pointless early end, is one of the great tragedies of mathematical history. Although Galois' group theory is considered one of the greatest developments of 19th century mathematics, Galois' writings were largely ignored until the revolutionary work of Klein and Lie.
James Joseph Sylvester (1814-1897) England, U.S.A.
Sylvester made important contributions in matrix theory, invariant theory, number theory, partition theory, reciprocant theory, geometry, and combinatorics. He invented the theory of elementary divisors, and co-invented the law of quadratic forms. It is said he coined more new mathematical terms (e.g. matrix, invariant, discriminant, covariant, syzygy, graph, Jacobian) than anyone except Leibniz. Sylvester was especially noted for the broad range of his mathematics and his ingenious methods. He solved (or partially solved) a huge variety of rich puzzles including various geometric gems; the enumeration of polynomial roots first tackled by Descartes and Newton; and, by advancing the theory of partitions, the system of equations posed by Euler as The Problem of the Virgins. Sylvester was also a linguist, a poet, and did work in mechanics (inventing the skew pantograph) and optics. He once wrote, "May not music be described as the mathematics of the sense, mathematics as music of the reason?"
Karl Wilhelm Theodor Weierstrass (1815-1897) Germany
Weierstrass devised new definitions for the primitives of calculus, developed the concept of uniform convergence, and was then able to prove several fundamental but hitherto unproven theorems. Starting strictly from the integers, he also applied his axiomatic methods to a definition of irrational numbers. He developed important new insights in other fields including the calculus of variations, elliptic functions, and trigonometry. Weierstrass shocked his colleagues when he demonstrated a continuous function which is differentiable nowhere. (Both this and the Bolzano-Weierstrass Theorem were rediscoveries of forgotten results by the under-published Bolzano.) He found simpler proofs of many existing theorems, including Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann Transcendence Theorem. Steiner's proof of the Isoperimetric Theorem contained a flaw, so Weierstrass became the first to supply a fully rigorous proof of that famous and ancient result. Peter Dirichlet was a champion of rigor, but Weierstrass discovered a flaw in the argument for Dirichlet's Principle of of variational calculus.
Weierstrass demonstrated extreme brilliance as a youth, but during his college years he detoured into drinking and dueling and ended up as a degreeless secondary school teacher. During this time he studied Abel's papers, developed results in elliptic and Abelian functions, proved the Laurent expansion theorem before Laurent did, and independently proved the Fundamental Theorem of Functions of a Complex Variable. He was interested in power series and felt that others had overlooked the importance of Abel's Theorem. Eventually one of his papers was published in a journal; he was immediately given an honorary doctorate and was soon regarded as one of the best and most inspirational mathematicians in the world. His insistence on absolutely rigorous proofs equaled or exceeded even that of Cauchy, Abel and Dirichlet. His students included Kovalevskaya, Frobenius, Mittag-Leffler, and several other famous mathematicians. Bell called him "probably the greatest mathematical teacher of all time." In 1873 Hermite called Weierstrass "the Master of all of us." Today he is often called the "Father of Modern Analysis."
Weierstrass once wrote: "A mathematician who is not also something of a poet will never be a complete mathematician."
George Boole (1815-1864) England
George Boole was a precocious child who impressed by teaching himself classical languages, but was too poor to attend college and became an elementary school teacher at age 16. He gradually developed his math skills; as a young man he published a paper on the calculus of variations, and soon became one of the most respected mathematicians in England despite having no formal training. He was noted for work in symbolic logic, algebra and analysis, and also was apparently the first to discover invariant theory. When he followed up Augustus de Morgan's earlier work in symbolic logic, de Morgan insisted that Boole was the true master of that field, and begged his friend to finally study mathematics at university. Boole couldn't afford to, and had to be appointed Professor instead!
Although very few recognized its importance at the time, it is Boole's work in Boolean algebra and symbolic logic for which he is now remembered; this work inspired computer scientists like Claude Shannon. Boole's book An Investigation of the Laws of Thought prompted Bertrand Russell to label him the "discoverer of pure mathematics."
Boole once said "No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful."
Pafnuti Lvovich Chebyshev (1821-1894) Russia
Pafnuti Chebyshev (Pafnuty Tschebyscheff) was noted for work in probability, number theory, approximation theory, integrals, the theory of equations, and orthogonal polynomials. His famous theorems cover a diverse range; they include a new version of the Law of Large Numbers, first rigorous proof of the Central Limit Theorem, and an important result in integration of radicals first conjectured by Abel. He invented the Chebyshev polynomials, which have very wide application; many other theorems or concepts are also named after him. He did very important work with prime numbers, proving that there is always a prime between any n and 2n, and working with the zeta function before Riemann did. He made much progress with the Prime Number Theorem, proving two distinct forms of that theorem, each incomplete but in a different way. Chebyshev was very influential for Russian mathematics, inspiring Andrei Markov and Aleksandr Lyapunov among others.
Chebyshev was also a premier applied mathematician and a renowned inventor; his several inventions include the Chebyshev linkage, a mechanical device to convert rotational motion to straight-line motion. He once wrote "To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls."
Arthur Cayley (1821-1895) England
Cayley was one of the most prolific mathematicians in history; a list of the branches of mathematics he pioneered will seem like an exaggeration. In addition to being very inventive, he was an excellent algorist; some considered him to be the greatest mathematician of the late 19th century (an era that includes Weierstrass and Poincaré). Cayley was the essential founder of modern group theory, matrix algebra, the theory of higher singularities, and higher-dimensional geometry (building on Plücker's work and anticipating the ideas of Klein), as well as the theory of invariants. Among his many important theorems are the Cayley-Hamilton Theorem, and Cayley's Theorem itself (that any group is isomorphic to a subgroup of a symmetric group). He extended Hamilton's quaternions and developed the octonions, but was still one of the first to realize that these special algebras could often be subsumed by general matrix methods. (Hamilton's friend John T. Graves independently discovered the octonions about the same time as Cayley did.) He also did original research in combinatorics (e.g. enumeration of trees), elliptic and Abelian functions, and projective geometry. One of his famous geometric theorems is a generalization of Pascal's Mystic Hexagram result; another resulted in an elegant proof of the Quadratic Reciprocity law.
Cayley may have been the least eccentric of the great mathematicians: In addition to his life-long love of mathematics, he enjoyed hiking, painting, reading fiction, and had a happy married life. He easily won Smith's Prize and Senior Wrangler at Cambridge, but then worked as a lawyer for many years. He later became professor, and finished his career in the limelight as President of the British Association for the Advancement of Science. He and James Joseph Sylvester were a source of inspiration to each other. These two, along with Charles Hermite, are considered the founders of the important theory of invariants. Though applied first to algebra, the notion of invariants is useful in many areas of mathematics.
Cayley once wrote: "As for everything else, so for a mathematical theory: beauty can be perceived but not explained."
Charles Hermite (1822-1901) France
Hermite studied the works of Lagrange and Gauss from an early age and soon developed an alternate proof of Abel's famous quintic impossibility result. He attended the same college as Galois and also had trouble passing their examinations, but soon became highly respected by Europe's best mathematicians for his significant advances in analytic number theory, elliptic functions, and quadratic forms. Along with Cayley and Sylvester, he founded the important theory of invariants. Hermite's theory of transformation allowed him to connect analysis, algebra and number theory in novel ways. He was a kindly modest man and an inspirational teacher. Among his students was Poincaré, who said of Hermite, "He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.... Methods always seemed to be born in his mind in some mysterious way." Hermite's other famous students included Darboux, Borel, and Hadamard who wrote of "how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being."
Although he and Abel had proved that the general quintic lacked algebraic solutions, Hermite introduced an elliptic analog to the circular trigonometric functions and used these to provide a general solution for the quintic equation. He developed the concept of complex conjugate which is now ubiquitous in mathematical physics and matrix theory. He was first to prove that the Stirling and Euler generalizations of the factorial function are equivalent. He was first to note remarkable facts about Heegner numbers, e.g.
eπ√163 = 262537412640768743.9999999999992...
(Without computers he was able to calculate this number, including the twelve 9's to the right of the decimal point.) Very many elegant concepts and theorems are named after Hermite. Hermite's most famous result may be his intricate proof that e (along with a broad class of related numbers) is transcendental. (Extending the proof to π was left to Lindemann, a matter of regret for historians, some of whom who regard Hermite as the greatest mathematician of his era.)
Ferdinand Gotthold Max Eisenstein (1823-1852) Germany
Eisenstein was born into severe poverty and suffered health problems throughout his short life, but was still one of the more significant mathematicians of his era. Today's mathematicians who study Eisenstein are invariably amazed by his brilliance and originality. He made revolutionary advances in number theory, algebra and analysis, and was also a composer of music. He anticipated ring theory, developed a new basis for elliptic functions, studied ternary quadratic forms, proved several theorems about cubic and higher-degree reciprocity, discovered the notion of analytic covariant, and much more.
Eisenstein was a young prodigy; he once wrote "As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork." Despite his early death, he is considered one of the greatest number theorists ever. Gauss named Eisenstein, along with Newton and Archimedes, as one of the three epoch-making mathematicians of history.
Leopold Kronecker (1823-1891) Germany
Kronecker was a businessman who pursued mathematics mainly as a hobby, but was still very prolific, and one of the greatest theorem provers of his era. He explored a wide variety of mathematics -- number theory, algebra, analysis, matrixes -- and especially the interconnections between areas. Many concepts and theorems are named after Kronecker; some of his theorems are frequently used as lemmas in algebraic number theory, ergodic theory, and approximation theory. He provided key ideas about foundations and continuity despite that he had philosophic objections to irrational numbers and infinities. He also introduced the Theory of Divisors to avoid Dedekind's Ideals; the importance of this and other work was only realized long after his death. Kronecker's philosophy eventually led to the Constructivism and Intuitionism of Brouwer and Poincaré.
Georg Friedrich Bernhard Riemann (1826-1866) Germany
Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He made revolutionary advances in complex analysis, which he connected to both topology and number theory. He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He introduced the Riemann integral which clarified analysis. He developed the theory of manifolds, a term which he invented. Manifolds underpin topology. By imposing metrics on manifolds Riemann invented differential geometry and took non-Euclidean geometry far beyond his predecessors. Riemann's other masterpieces include tensor analysis, the theory of functions, and a key relationship between some differential equation solutions and hypergeometric series. His generalized notions of distance and curvature described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch Theorem, a key connection among topology, complex analysis and algebraic geometry. He proved Riemann's Rearrangement Theorem, a strong (and paradoxical) result about conditionally convergent series. He was also first to prove theorems named after others, e.g. Green's Theorem. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's curvature tensor and other notions of the geometry of space.
Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous lecture Riemann went far beyond Gauss' initial effort in differential geometry, extended it to multiple dimensions, and introduced the new and important theory of differential manifolds. Five years later, to celebrate his election to the Berlin Academy, Riemann presented a lecture "On the Number of Prime Numbers Less Than a Given Quantity," for which "Number" he presented and partially proved an exact formula, albeit weirdly complicated. Numerous papers have been written on the distribution of primes, but Riemann's contribution is incomparable, despite that his Berlin Academy lecture was his only paper ever on the topic, and number theory was far from his specialty. In the lecture he posed the Hypothesis of Riemann's zeta function, needed for the missing step in his proof. This Hypothesis is considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.") ζ(.) was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions of ζ(s = a+bi) = 0, either s has real part a=1/2 or imaginary part b=0.
Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality;" another biographer called him "one of the most profound and imaginative mathematicians of all time [and] a great philosopher"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."
Henry John Stephen Smith (1826-1883) England
Henry Smith (born in Ireland) was one of the greatest number theorists, working especially with elementary divisors; he also advanced the theory of quadratic forms. A famous problem of Eisenstein was, given n and k, in how many different ways can n be expressed as the sum of k squares? Smith made great progress on this problem, subsuming special cases which had earlier been famous theorems. Although most noted for number theory, he had great breadth. He did prize-winning work in geometry, discovered the unique normal form for matrices which now bears his name, anticipated specific fractals including the Cantor set, the Sierpinski gasket and the Koch snowflake, and wrote a paper demonstrating the limitations of Riemann integration.
Smith is sometimes called "the mathematician the world forgot." His paper on integration could have led directly to measure theory and Lebesgue integration, but was ignored for decades. The fractals he discovered are named after people who rediscovered them. The Smith-Minkowski-Siegel mass formula of lattice theory would be called just the Smith formula, but had to be rediscovered. And his solution to the Eisenstein five-squares problem, buried in his voluminous writings on number theory, was ignored: this "unsolved" problem was featured for a prize which Minkowski won two decades later!
Henry Smith was an outstanding intellect with a modest and charming personality. He was knowledgeable in a broad range of fields unrelated to mathematics; his University even insisted he run for Parliament. His love of mathematics didn't depend on utility: he once wrote "Pure mathematics: may it never be of any use to anyone."
Antonio Luigi Gaudenzio Giuseppe Cremona (1830-1903) Italy
Luigi Cremona made many important advances in analytic, synthetic and projective geometry, especially in the transformations of algebraic curves and surfaces. Working in mathematical physics, he developed the new field of graphical statics, and used it to reinterpret some of Maxwell's results. He improved (or found brilliant proofs for) several results of Steiner, especially in the field of cubic surfaces. (Some of this work was done in collaboration with Rudolf Sturm.) He is especially noted for developing the theory of Cremona transformations which have very wide application. He found a generalization of Pascal's Mystic Hexagram. Cremona also played a political role in establishing the modern Italian state and, as an excellent teacher, helped make Italy a top center of mathematics.
James Clerk Maxwell (1831-1879) Scotland
At the age of 14, Maxwell published a remarkable paper on the construction of ovals; these were an independent discovery of the Ovals of Descartes, but Maxwell allowed more than two foci, had elaborate configurations (he was drawing the ovals with string and pencil), and identified errors in Descartes' treatment of them. His genius was soon renowned throughout Scotland, with the future Lord Kelvin remarking that Maxwell's "lively imagination started so many hares that before he had run one down he was off on another." He did a comprehensive analysis of Saturn's rings; developed the important kinetic theory of gases; explored elasticity, viscosity, knot theory, topology, soap bubbles, and more. He introduced the "Maxwell's Demon" as a thought experiment for thermodynamics; his paper "On Governors" effectively founded the field of cybernetics; he advanced the theory of color, and produced the first color photograph. One Professor said of him, "there is scarcely a single topic that he touched upon, which he did not change almost beyond recognition." Maxwell was also a poet.
Maxwell did little of importance in pure mathematics, so his great creativity in mathematical physics might not seem enough to qualify him for this list, although his contribution to the kinetic theory of gases (which even led to the first estimate of molecular sizes) would already be enough to make him one of the greatest physicists. But then, in 1864 James Clerk Maxwell stunned the world by publishing the equations of electricity and magnetism, predicting the existence of radio waves and that light itself is a form of such waves and is thus linked to the electro-magnetic force. Richard Feynman considered this the most significant event of the 19th century (though others might give higher billing to Darwin's theory of evolution). Along with Einstein, Newton, Galileo and Archimedes, Maxwell would be the near-certain choice for a Five Greatest Physicists list. Recalling Newton's comment about "standing on the shoulders" of earlier greats, Einstein was asked whose shoulders he stood on; he didn't name Newton: he said "Maxwell." Maxwell has been called the "Father of Modern Physics"; he ranks #24 on Hart's list of the Most Influential Persons in History. (Faraday, the great experimentalist who noted the electromagnetic effects which Maxwell later rendered in mathematics, ranks #23 on Hart's list.)
Julius Wilhelm Richard Dedekind (1831-1916) Germany
Dedekind was one of the most innovative mathematicians ever; his clear expositions and rigorous axiomatic methods had great influence. He made seminal contributions to abstract algebra and algebraic number theory as well as mathematical foundations. He was one of the first to pursue Galois Theory, making major advances there and pioneering in the application of group theory to other branches of mathematics. Dedekind also invented a system of fundamental axioms for arithmetic, worked in probability theory and complex analysis, and invented prime partitions and modular lattices. Dedekind may be most famous for his theory of ideals and rings; Kronecker and Kummer had begun this, but Dedekind gave it a more abstract and productive basis, which was developed further by Hilbert, Noether and Weil. Though the term ring itself was coined by Hilbert, Dedekind introduced the terms module, field, and ideal. Dedekind was far ahead of his time, so Noether became famous as the creator of modern algebra; but she acknowledged her great predecessor, frequently saying "It is all already in Dedekind."
Dedekind was concerned with rigor, writing "nothing capable of proof ought to be accepted without proof." Before him, the real numbers, continuity, and infinity all lacked rigorous definitions. The axioms Dedekind invented allow the integers and rational numbers to be built and his Dedekind Cut then led to a rigorous and useful definition of the real numbers. Dedekind was a key mentor for Georg Cantor: he introduced the notion that a bijection implied equinumerosity, used this to define infinitude (a set is infinite if equinumerous with its proper subset), and was first to prove the Cantor-Bernstein-Schröder Theorem, though he didn't publish his proof. (Because he spent his career at a minor university, and neglected to publish some of his work, Dedekind's contributions may be underestimated.)
Rudolf Friedrich Alfred Clebsch (1833-1872) Germany
Alfred Clebsch began in mathematical physics, working in hydrodynamics and elasticity, but went on to become a pure mathematician of great brilliance and versatility. He started with novel results in analysis, but went on to make important advances to the invariant theory of Cayley and Sylvester (and Salmon and Aronhold), to the algebraic geometry and elliptic functions of Abel and Jacobi, and to the enumerative and projective geometries of Plücker. He was also one of the first to build on Riemann's innovations. Clebsch developed new notions, e.g. Clebsch-Aronhold symbolic notation and 'connex'; and proved key theorems about cubic surfaces (for example, the Sylvester pentahedron conjecture) and other high-degree curves, and representations (bijections) between surfaces. Some of his work, e.g. Clebsch-Gordan coefficients which are important in physics, was done in collaboration with Paul Gordan. For a while Clebsch was one of the top mathematicians in Germany, and founded an important journal, but he died young. He was a key teacher of Max Noether, Ferdinand Lindemann, Alexander Brill and Gottlob Frege. Clebsch's great influence is suggested by the fact that his name appeared as co-author on a text published 60 years after his death.
Eugenio Beltrami (1835-1899) Italy
Beltrami was an outstanding mathematician noted for early insights connecting geometry and topology (differential geometry, pseudospherical surfaces, etc.), transformation theory, differential calculus, and especially for proving the equiconsistency of hyperbolic and Euclidean geometry for every dimensionality; he achieved this by building on models of Cayley, Klein, Riemann and Liouville. He was first to invent singular value decompositions. (Camille Jordan and J.J. Sylvester each invented it independently a few years later.) Using insights from non-Euclidean geometry, he did important mathematical work in a very wide range of physics; for example he improved Green's theorem, generalized the Laplace operator, studied gravitation in non-Euclidean space, and gave a new derivation of Maxwell's equations.
Marie Ennemond Camille Jordan (1838-1921) France
Jordan was a great "universal mathematician", making revolutionary advances in group theory, topology, and operator theory; and also doing important work in differential equations, number theory, measure theory, matrix theory, combinatorics, algebra and especially Galois theory. He worked as both mechanical engineer and professor of analysis. Jordan is especially famous for the Jordan Closed Curve Theorem of topology, a simple statement "obviously true" yet remarkably difficult to prove. In measure theory he developed Peano-Jordan "content" and proved the Jordan Decomposition Theorem. He also proved the Jordan-Holder Theorem of group theory, invented the notion of homotopy, invented the Jordan Canonical Forms of matrix theory, and supplied the first complete proof of Euler's Polyhedral Theorem, F+V = E+2. Some consider Jordan second only to Weierstrass among great 19th-century teachers; his work inspired such mathematicians as Klein, Lie and Borel.
Marius Sophus Lie (1842-1899) Norway
Lie was twenty-five years old before his interest in and aptitude for mathematics became clear, but then did revolutionary work with continuous symmetry and continuous transformation groups. These groups and the algebra he developed to manipulate them now bear his name; they have major importance in the study of differential equations. Lie sphere geometry is one result of Lie's fertile approach and even led to a new approach for Apollonius' ancient problem about tangent circles. Lie became a close friend and collaborator of Felix Klein early in their careers; their methods of relating group theory to geometry were quite similar; but they eventually fell out after Klein became (unfairly?) recognized as the superior of the two. Lie's work wasn't properly appreciated in his own lifetime, but one later commentator was "overwhelmed by the richness and beauty of the geometric ideas flowing from Lie's work."
Jean Gaston Darboux (1842-1917) France
Darboux did outstanding work in geometry, differential geometry, analysis, function theory, mathematical physics, and other fields, his ability "based on a rare combination of geometrical fancy and analytical power." He devised the Darboux integral, equivalent to Riemann's integral but simpler; developed a novel mapping between (hyper-)sphere and (hyper-)plane; proved an important Envelope Theorem in the calculus of variations; developed the field of infinitesimal geometry; and more. Several important theorems are named after him including a generalization of Taylor series, the foundational theorem of symplectic geometry, and the fact that "the image of an interval is also an interval." He wrote the definitive textbook on differential geometry; he was an excellent teacher, inspiring Borel, Cartan and others.
William Kingdon Clifford (1845-1879) England
Clifford was a versatile and talented mathematician who was among the first to appreciate the work of both Riemann and Grassmann. He found new connections between algebra, topology and non-Euclidean geometry. Combining Hamilton's quaternions, Grassmann's exterior algebra, and his own geometric intuition and understanding of physics, he developed biquaternions, and generalized this to geometric algebra, which paralleled work by Klein. In addition to developing theories, he also produced ingenious proofs; for example he was first to prove Miquel's n-Circle Theorem, and did so with a purely geometric argument. Clifford is especially famous for anticipating, before Einstein, that gravitation could be modeled with a non-Euclidean space. He was a polymath; a talented teacher, noted philosopher, writer of children's fairy tales, and outstanding athlete. With his singular genius, Clifford would probably have become one of the greatest mathematicians of his era had he not died at age thirty-three.
Georg Cantor (1845-1918) Russia, Germany
Cantor did brilliant and important work early in his career, for example he greatly advanced the Fourier-series uniqueness question which had intrigued Riemann. In his explorations of that problem he was led to questions of set enumeration, and his greatest invention: set theory. Cantor created modern Set Theory almost single-handedly, defining cardinal numbers, well-ordering, ordinal numbers, and discovering the Theory of Transfinite Numbers. He defined equality between cardinal numbers based on the existence of a bijection, and was the first to demonstrate that the real numbers have a higher cardinal number than the integers. (He also showed that the rationals have the same cardinality as the integers; and that the reals have the same cardinality as the points of N-space and as the power-set of the integers.) Although there are infinitely many distinct transfinite numbers, Cantor conjectured that C, the cardinality of the reals, was the second smallest transfinite number. This Continuum Hypothesis was included in Hilbert's famous List of Problems, and was partly resolved many years later: Cantor's Continuum Hypothesis is an "Undecidable Statement" of Set Theory.
Cantor's revolutionary set theory attracted vehement opposition from Poincaré ("grave disease"), Kronecker (Cantor was a "charlatan" and "corrupter of youth"), Wittgenstein ("laughable nonsense"), and even theologians. David Hilbert had kinder words for it: "The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity" and addressed the critics with "no one shall expel us from the paradise that Cantor has created." Cantor's own attitude was expressed with "The essence of mathematics lies in its freedom." Cantor's set theory laid the theoretical basis for the measure theory developed by Borel and Lebesgue. Cantor's invention of modern set theory is now considered one of the most important and creative achievements in modern mathematics.
Cantor demonstrated much breadth (he even involved himself in the Shakespeare authorship controversy!). In addition to his set theory and key discoveries in the theory of trigonometric series, he made advances in number theory, and gave the modern definition of irrational numbers. His Cantor set was the early inspiration for fractals. Cantor was also an excellent violinist. He once wrote "In mathematics the art of proposing a question must be held of higher value than solving it."
Friedrich Ludwig Gottlob Frege (1848-1925) Germany
Gottlob Frege developed the first complete and fully rigorous system of pure logic; his work has been called the greatest advance in logic since Aristotle. He introduced the essential notion of quantifiers; he distinguished terms from predicates, and simple predicates from 2nd-level predicates. From his second-order logic he defined numbers, and derived the axioms of arithmetic with what is now called Frege's Theorem. His work was largely underappreciated at the time, partly because of his clumsy notation, partly because his system was published with a flaw (Russell's antinomy). (Bertrand Russell reports that when he informed him of this flaw, Frege took it with incomparable integrity, grace, and even intellectual pleasure.) Frege and Cantor were the era's outstanding foundational theorists; unfortunately their relationship with each other became bitter. Despite all this, Frege's work influenced Peano, Russell, Wittgenstein and others; and he is now often called the greatest mathematical logician ever.
Frege also did work in geometry and differential equations; and, in order to construct the real numbers with his set theory, proved an important new theorem of group theory. He was also an important philosopher, and an essential founder of "analytic philosophy." He wrote "Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician."
Ferdinand Georg Frobenius (1849-1917) Germany
Frobenius did significant work in a very broad range of mathematics, was an outstanding algorist, and had several successful students including Edmund Landau, Issai Schur, and Carl Siegel. In addition to developing the theory of abstract groups, Frobenius did important work in number theory, differential equations, elliptic functions, biquadratic forms, matrixes, and algebra. He was first to actually prove the important Cayley-Hamilton Theorem, and first to extend the Sylow Theorems to abstract groups. He anticipated the important and imaginative Prime Density Theorem, though he didn't prove its general case. He developed the method of Cesáro summation of divergent series before Cesáro did. Although he modestly left his name off the "Cayley-Hamilton Theorem," many lemmas and concepts are named after him, including Frobenius conjugacy class, Frobenius reciprocity, Frobenius manifolds, the Frobenius-Schur Indicator, etc. He is most noted for his character theory, a revolutionary advance which led to the representation theory of groups, and has applications in modern physics. The middle-aged Frobenius invented this after the aging Dedekind asked him for help in solving a key algebraic factoring problem.
Christian Felix Klein (1849-1925) Germany
Klein's key contribution was an application of invariant theory to unify geometry with group theory. This radical new view of geometry inspired Sophus Lie's Lie groups, and also led to the remarkable unification of Euclidean and non-Euclidean geometries which is probably Klein's most famous result. Klein did other work in function theory, providing links between several areas of mathematics including number theory, group theory, hyperbolic geometry, and abstract algebra. His Klein's Quartic curve and popularly-famous Klein's bottle were among several useful results from his new approaches to groups and higher-dimensional geometries and equations. Klein did significant work in mathematical physics, e.g. writing about gyroscopes. He facilitated David Hilbert's early career, publishing his controversial Finite Basis Theorem and declaring it "without doubt the most important work on general algebra [the leading German journal] ever published."
Klein is also famous for his book on the icosahedron, reasoning from its symmetries to develop the elliptic modular and automorphic functions which he used to solve the general quintic equation. He formulated a "grand uniformization theorem" about automorphic functions but suffered a health collapse before completing the proof. His focus then changed to teaching; he devised a mathematics curriculum for secondary schools which had world-wide influence. Klein once wrote "... mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs."
Oliver Heaviside (1850-1925) England
Heaviside dropped out of high school to teach himself telegraphy and electromagnetism, becoming first a telegraph operator but eventually perhaps the greatest electrical engineer ever. He developed transmission line theory, invented the coaxial cable, predicted Cherenkov radiation, described the use of the ionosphere in radio transmission, and much more. Some of his insights anticipated parts of special relativity, and he was first to speculate about gravitational waves. For his revolutionary discoveries in electromagnetism and mathematics, Heaviside became the first winner of the Faraday Medal.
As an applied mathematician, Heaviside developed operational calculus (an important shortcut for solving differential equations); developed vector analysis independently of Grassmann; and demonstrated the usage of complex numbers for electro-magnetic equations. Four of the famous Maxwell's Equations are in fact due to Oliver Heaviside, Maxwell having presented a more cumbersome version. Although one of the greatest applied mathematicians, Heaviside is omitted from the Top 100 because he didn't provide proofs for his methods. Of this Heaviside said, "Should I refuse a good dinner simply because I do not understand the process of digestion?"
Sofia Vasilyevna Kovalevskaya (1850-1891) Russia
Sofia Kovalevskaya (aka Sonya Kowalevski; née Korvin-Krukovskaya) was initially self-taught, sought out Weierstrass as her teacher, and was later considered the greatest female mathematician ever (before Emmy Noether). She was influential in the development of Russian mathematics. Kovalevskaya studied Abelian integrals and partial differential equations, producing the important Cauchy-Kovalevsky Theorem; her application of complex analysis to physics inspired Poincaré and others. Her most famous work was the solution to the Kovalevskaya top, which has been called a "genuine highlight of 19th-century mathematics." Other than the simplest cases solved by Euler and Lagrange, exact ("integrable") solutions to the equations of motion were unknown, so Kovalevskaya received fame and a rich prize when she solved the Kovalevskaya top. Her ingenious solution might be considered a mere curiosity, but since it is still the only post-Lagrange physical motion problem for which an "integrable" solution has been demonstrated, it remains an important textbook example. Kovalevskaya once wrote "It is impossible to be a mathematician without being a poet in soul." She was also a noted playwright.
Jules Henri Poincaré (1854-1912) France
Poincaré founded the theory of algebraic (combinatorial) topology, and is sometimes called the "Father of Topology" (a title also used for Euler and Brouwer). He also did brilliant work in several other areas of mathematics; he was one of the most creative mathematicians ever, and the greatest mathematician of the Constructivist ("intuitionist") style. He published hundreds of papers on a variety of topics and might have become the most prolific mathematician ever, but he died at the height of his powers. Poincaré was clumsy and absent-minded; like Galois, he was almost denied admission to French University, passing only because at age 17 he was already far too famous to flunk.
In addition to his topology, Poincaré laid the foundations of homology; he discovered automorphic functions (a unifying foundation for the trigonometric and elliptic functions), and essentially founded the theory of periodic orbits; he made major advances in the theory of differential equations. He is credited with partial solution of Hilbert's 22nd Problem. Several important results carry his name, for example the famous Poincaré Recurrence Theorem, which almost seems to contradict the Second Law of Thermodynamics. Poincaré is especially noted for effectively discovering chaos theory, and for posing Poincaré's Conjecture; that conjecture was one of the most famous unsolved problems in mathematics for an entire century, and can be explained without equations to a layman. The Conjecture is that all "simply-connected" closed manifolds are topologically equivalent to "spheres"; it is directly relevant to the possible topology of our universe. Recently Grigori Perelman proved Poincaré's conjecture, and is eligible for the first Million Dollar math prize in history.
As were most of the greatest mathematicians, Poincaré was intensely interested in physics. He made revolutionary advances in fluid dynamics and celestial motions; he anticipated Minkowski space and much of Einstein's Special Theory of Relativity (including the famous equation E = mc2). Poincaré also found time to become a famous popular writer of philosophy, writing "Mathematics is the art of giving the same name to different things;" and "A [worthy] mathematician experiences in his work the same impression as an artist; his pleasure is as great and of the same nature;" and "If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living." With his fame, Poincaré helped the world recognize the importance of the new physical theories of Einstein and Planck.
Andrei Andreyevich Markov (1856-1922) Russia
Markov did excellent work in a broad range of mathematics including analysis, number theory, algebra, continued fractions, approximation theory, and especially probability theory: it has been said that his accuracy and clarity transformed probability theory into one of the most perfected areas of mathematics. Markov is best known as the founder of the theory of stochastic processes. In addition to his Ergodic Theorem about such processes, theorems named after him include the Gauss-Markov Theorem of statistics, the Riesz-Markov Theorem of functional analysis, and the Markov Brothers' Inequality in the theory of equations. Markov was also noted for his politics, mocking Czarist rule, and insisting that he be excommunicated from the Russian Orthodox Church when Tolstoy was.
Markov had a son, also named Andrei Andreyevich, who was also an outstanding mathematician of great breadth. Among the son's achievements was Markov's Theorem, which helps relate the theories of braids and knots to each other.
Giuseppe Peano (1858-1932) Italy
Giuseppe Peano is one of the most under-appreciated of all great mathematicians. He started his career by proving a fundamental theorem in differential equations, developed practical solution methods for such equations, discovered a continuous space-filling curve (then thought impossible), and laid the foundations of abstract operator theory. He also produced the best calculus textbook of his time, was first to produce a correct (non-paradoxical) definition of surface area, proved an important theorem about Dirichlet functions, did important work in topology, and much more. Much of his work was unappreciated and left for others to rediscover: he anticipated many of Borel's and Lebesgue's results in measure theory, and several concepts and theorems of analysis. He was the champion of counter-examples, and found flaws in published proofs of several important theorems.
Most of the preceding work was done when Peano was quite young. Later he focused on mathematical foundations, and this is the work for which he is most famous. He developed rigorous definitions and axioms for set theory, as well as most of the notation of modern set theory. He was first to define arithmetic (and then the rest of mathematics) in terms of set theory. Peano was first to note that some proofs required an explicit Axiom of Choice (although it was Ernst Zermelo who explicitly formulated that Axiom a few years later).
Despite his early show of genius, Peano's quest for utter rigor may have detracted from his influence in mainstream mathematics. Moreover, since he modestly referenced work by predecessors like Dedekind, Peano's huge influence in axiomatic theory is often overlooked. Yet Bertrand Russell reports that it was from Peano that he first learned that a single-member set is not the same as its element; this fact is now taught in elementary school.