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Added by vadeboncoeur on 5 Sep 2016 02:54
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Most famous mathematicians

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Aristotle is considered the greatest scientist of the ancient world, and the most influential philosopher and logician ever; he ranks #13 on Michael Hart's list of the Most Influential Persons in History. (His science was a standard curriculum for almost 2000 years, unfortunate since many of his ideas were quite mistaken.) He was personal tutor to the young Alexander the Great. Aristotle's writings on definitions, axioms and proofs may have influenced Euclid. He was also the first mathematician to write on the subject of infinity. His writings include geometric theorems, some with proofs different from Euclid's or missing from Euclid altogether; one of these (which is seen only in Aristotle's work prior to Apollonius) is that a circle is the locus of points whose distances from two given points are in constant ratio. Even if, as is widely agreed, Aristotle's geometric theorems were not his own work, his status as the most influential logician and philosopher makes him a candidate for the List.
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Descartes' early career was that of soldier-adventurer and he finished as tutor to royalty, but in between he achieved fame as the preeminent intellectual of his day. He is considered the inventor of both analytic geometry and symbolic algebraic notation and is therefore called the "Father of Modern Mathematics." His use of equations to partially solve the geometric Problem of Pappus revolutionized mathematics. Because of his famous philosophical writings ("Cogito ergo sum") he is considered, along with Aristotle, to be one of the most influential thinkers in history. He ranks #49 on Michael Hart's famous list of the Most Influential Persons in History. His famous mathematical theorems include the Rule of Signs (for determining the signs of polynomial roots), the elegant formula relating the radii of Soddy kissing circles, his theorem on total angular defect (an early form of the Gauss-Bonnet result so key to much mathematics), and an improved solution to the Delian problem (cube-doubling). He improved mathematical notation (e.g. the use of superscripts to denote exponents). He also discovered Euler's Polyhedral Theorem, F+V = E+2. Descartes was very influential in physics and biology as well, e.g. developing laws of motion which included a "vortex" theory of gravitation; but most of his scientific work outside mathematics was eventually found to be incorrect.
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Albert Einstein was unquestionably one of the two greatest physicists in all of history. The atomic theory achieved general acceptance only after Einstein's 1905 paper which showed that atoms' discreteness explained Brownian motion. Another 1905 paper introduced the famous equation E = mc2; yet Einstein published other papers that same year, two of which were more important and influential than either of the two just mentioned. No wonder that physicists speak of the Miracle Year without bothering to qualify it as Einstein's Miracle Year! (Before his Miracle Year, Einstein had been a mediocre undergraduate, and held minor jobs including patent examiner.) Altogether Einstein published at least 300 books or papers on physics. For example, in a 1917 paper he anticipated the principle of the laser. Also, he was co-inventor of several devices, including a gyroscopic compass, hearing aid, automatic camera and, most famously, the Einstein-Szilard refrigerator. He became a very famous and influential public figure. (For example, it was his letter that led Roosevelt to start the Manhattan Project.) Among his many famous quotations is: "The search for truth is more precious than its possession."
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Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe's. Descriptions which have been applied to Leibniz include "one of the two greatest universal geniuses" (da Vinci was the other); "the most important logician between Aristotle and Boole;" and the "Father of Applied Science." Leibniz described himself as "the most teachable of mortals."
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Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. However Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inverse-square gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") Newton published the Cooling Law of thermodynamics. He also made contributions to chemistry, and was the important early advocate of the atomic theory. His writings also made important contributions to the general scientific method. (His other intellectual interests included theology, astrology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the huge range of his physics: even without his Laws of Motion, Gravitation and Cooling, he'd be famous just for his revolutionary work in optics, where he explained diffraction, observed that white light is a mixture of all the rainbow's colors, noted that purple is created by combining red and blue light and, starting from that observation, was first to conceive of a color hue "wheel." Newton almost anticipated Einstein's mass-energy equivalence, writing "Gross Bodies and Light are convertible into one another... [Nature] seems delighted with Transmutations." Newton's earliest fame came when he designed the first reflecting telescope: by avoiding chromatic aberration, these were the best telescopes of that era. He also designed the first reflecting microscope, and the sextant.
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Galileo discovered the laws of inertia, falling bodies (including parabolic trajectories), and the pendulum; he also introduced the notion of relativity which Einstein later found so fruitful. He was a great inventor: in addition to being first to conceive of the pendulum clock, he developed a new type of pump, and the best telescope, thermometer, hydrostatic balance, and cannon sector of his day. As a famous astronomer, Galileo pointed out that Jupiter's Moons, which he discovered, provide a natural clock and allow a universal time to be determined by telescope anywhere on Earth. (This was of little use in ocean navigation since a ship's rocking prevents the required delicate observations.) His discovery that Venus, like the Moon, had phases was the critical fact which forced acceptance of Copernican heliocentrism. His contributions outside physics and astronomy were also enormous: He invented the compound microscope and made early discoveries with it. He also made very important contributions to the early development of biology; but perhaps Galileo's most important contribution was to postulate universal laws of mechanics, in contrast to Aristotelian and religious notions of separate laws for heaven and earth.
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Pascal was an outstanding genius who studied geometry as a child. At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the "Cat's Cradle" or the "Mystic Hexagram." Pascal followed up this result by showing that each of Apollonius' famous theorems about conic sections was a corollary of the Mystic Hexagram; along with Gérard Desargues (1591-1661), he was a key pioneer of projective geometry. He also made important early contributions to calculus; indeed it was his writings that inspired Leibniz. Returning to geometry late in life, Pascal advanced the theory of the cycloid. In addition to his work in geometry and calculus, he founded probability theory, and made contributions to axiomatic theory. His name is associated with the Pascal's Triangle of combinatorics and Pascal's Wager in theology.
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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). He 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.
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Johannes Kepler
Kepler was interested in astronomy from an early age, studied to become a Lutheran minister, became a professor of mathematics instead, then Tycho Brahe's understudy, and, on Brahe's death, was appointed Imperial Mathematician at the age of twenty-nine. His observations of the planets with Brahe, along with his study of Apollonius' 1800-year old work, led to Kepler's three Laws of Planetary Motion, which in turn led directly to Newton's Laws of Motion. Beyond his discovery of these Laws (one of the most important achievements in all of science), Kepler is also sometimes called the "Founder of Modern Optics." He furthered the theory of the camera obscura, and was first to study the operation of the human eye, telescopes built from two convex lenses, and atmospheric refraction. Kepler was first to explain tides correctly. (Galileo dismissed this as well as Kepler's elliptical orbits, and later published his own incorrect explanation of tides.) Kepler ranks #75 on Michael Hart's famous list of the Most Influential Persons in History. This rank, much lower than that of Copernicus, Galileo or Newton, seems to me to underestimate Kepler's importance, since it was Kepler's Laws, rather than just heliocentrism, which were essential to the early development of mathematical physics.
Euler may be the most influential mathematician who ever lived (though some would make him second to Euclid); he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His colleagues called him "Analysis Incarnate." Laplace, famous for denying credit to fellow mathematicians, once said "Read Euler: he is our master in everything." His notations and methods in many areas are in use to this day. Euler was the most prolific mathematician in history and is often judged to be the best algorist of all time. (This brief summary can only touch on a few highlights of Euler's work. The ranking #4 may seem too low for this supreme mathematician, but Gauss succeeded at proving several theorems which had stumped Euler.)
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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.
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Jean le Rond d'Alembert
During the century after Newton, the Laws of Motion needed to be clarified and augmented with mathematical techniques. Jean le Rond, named after the Parisian church where he was abandoned as a baby, played a very key role in that development. His D'Alembert's Principle clarified Newton's Third Law and allowed problems in dynamics to be expressed with simple partial differential equations; his Method of Characteristics then reduced those equations to ordinary differential equations; to solve the resultant linear systems, he effectively invented the method of eigenvalues; he also anticipated the Cauchy-Riemann Equations. These are the same techniques in use for many problems in physics to this day. D'Alembert was also a forerunner in functions of a complex variable, and the notions of infinitesimals and limits. With his treatises on dynamics, elastic collisions, hydrodynamics, cause of winds, vibrating strings, celestial motions, refraction, etc., the young Jean le Rond easily surpassed the efforts of his older rival, Daniel Bernoulli. He may have been first to speak of time as a "fourth dimension." (Rivalry with the Swiss mathematicians led to d'Alembert's sometimes being unfairly ridiculed, although it does seem true that d'Alembert had very incorrect notions of probability.)
Euclid of Alexandria
Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers; he stated and proved the Unique Factorization Theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) His books contain many famous theorems, though many are attributed to predecessors like Hippocrates, Theodorus, Eudoxus, Archytas and Theaetetus. He may have proved that rigid-compass constructions can be implemented with collapsing-compass constructions. Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.
Carl Friedrich Gauss
Carl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three. His revolutionary nature was demonstrated at age twelve, when he began questioning the axioms of Euclid. His genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible if and only if it is the product of distinct prime Fermat numbers. Also at age 19, he proved Fermat's conjecture that every number is the sum of three triangle numbers. (He further determined the number of distinct ways such a sum could be formed.) At age 24 he published Disquisitiones Arithmeticae, probably the greatest book of pure mathematics ever.
James Clerk Maxwell
Maxwell published a remarkable paper on the construction of novel ovals, at the age of 14; 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, knot theory, 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.
Kurt Gödel
Gödel, who had the nickname Herr Warum ("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic. He partially resolved both Hilbert's 1st and 2nd Problems, the latter with a proof so remarkable that it was connected to the drawings of Escher and music of Bach in the title of a famous book. He was a close friend of Albert Einstein, and was first to discover "paradoxical" solutions (e.g. time travel) to Einstein's equations. About his friend, Einstein later said that he had remained at Princeton's Institute for Advanced Study merely "to have the privilege of walking home with Gödel." (Like a few of the other greatest 20th-century mathematicians, Gödel was very eccentric.)
John von Neumann
John von Neumann (born Neumann Janos Lajos) was a childhood prodigy who could do very complicated mental arithmetic at an early age. As an adult he was noted for hedonism and reckless driving but also became one of the most prolific geniuses in history, making major contributions in many branches of both pure and applied mathematics. He was an essential pioneer of both quantum physics and computer science.
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Turing developed a new foundation for mathematics based on computation; he invented the abstract Turing machine, designed a "universal" version of such a machine, proved the famous Halting Theorem (related to Godel's Incompleteness Theorem), and developed the concept of machine intelligence (including his famous Turing Test proposal). He also introduced the notions of definable number and oracle (important in modern computer science), and was an early pioneer in the study of neural networks. For this work he is called the Father of Computer Science and Artificial Intelligence. Turing also worked in group theory, numerical analysis, and complex analysis; he developed an important theorem about Riemann's zeta function; he had novel insights in quantum physics. During World War II he turned his talents to cryptology; his creative algorithms were considered possibly "indispensable" to the decryption of German Naval Enigma coding, which in turn is judged to have certainly shortened the War by at least two years. Although his clever code-breaking algorithms were his most spectacular contributions at Bletchley Park, he was also a key designer of the Bletchley "Bombe" computer. After the war he helped design other physical computers, as well as theoretical designs; and helped inspire von Neumann's later work. He (and earlier, von Neumann) wrote about the Quantum Zeno Effect which is sometimes called the Turing Paradox. He also studied the mathematics of biology, especially the Turing Patterns of morphogenesis which anticipated the discovery of BZ reactions. Turing's life ended tragically: charged with immorality and forced to undergo chemical castration, he apparently took his own life. With his outstanding depth and breadth, Alan Turing would qualify for our list in any event, but his decisive contribution to the war against Hitler gives him unusually strong historic importance.
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Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed, and even leapfrogged, the works of Euclid. (For example, some of Euclid's more difficult theorems are easy analytic consequences of Archimedes' Lemma of Centroids.) His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequaled clarity, with a modern biographer (Heath) describing Archimedes' treatises as "without exception monuments of mathematical exposition ... so impressive in their perfection as to create a feeling akin to awe in the mind of the reader." Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry. He was first to prove Heron's formula for the area of a triangle. His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He developed a recursive method of representing large integers, and was first to note the law of exponents, 10a·10b = 10a+b. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Some of Archimedes' work survives only because Thabit ibn Qurra translated the otherwise-lost Book of Lemmas; it contains the angle-trisection method and several ingenious theorems about inscribed circles. (Thabit shows how to construct a regular heptagon; it may not be clear whether this came from Archimedes, or was fashioned by Thabit by studying Archimedes' angle-trisection method.) Other discoveries known only second-hand include the Archimedean semiregular solids reported by Pappus, and the Broken-Chord Theorem reported by Alberuni.
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David Hilbert
Hilbert, often considered the greatest mathematician of the 20th century, was unequaled in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. He proved many new theorems, including the fundamental theorems of algebraic manifolds, and also discovered simpler proofs for older theorems. His examination of calculus led him to the invention of Hilbert space, considered one of the key concepts of functional analysis and modern mathematical physics. His Nullstellensatz Theorem laid the foundation of algebraic geometry. He was a founder of fields like metamathematics and modern logic. He was also the founder of the "Formalist" school which opposed the "Intuitionism" of Kronecker and Brouwer. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his Finite Basis Theorem, now regarded as one of the most important results of general algebra. His mentor, Paul Gordan, had sought the proof for many years, and rejected Hilbert's proof as non-constructive. Later, Hilbert produced the first constructive proof of the Finite Basis Theorem, as well. In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring Theorem.
Hermann Weyl
Weyl studied under Hilbert and became one of the premier mathematicians of the 20th century. His discovery of gauge invariance and notion of Riemann surfaces form the basis of modern physics. He excelled at many fields including integral equations, harmonic analysis, analytic number theory, Diophantine approximations, and the foundations of mathematics, but he is most respected for his revolutionary advances in geometric function theory (e.g., differentiable manifolds), the theory of compact groups (incl. representation theory), and theoretical physics (e.g., Weyl tensor, gauge field theory and invariance). For a while, Weyl was a disciple of Brouwer's Intuitionism and helped advance that doctrine, but he eventually found it too restrictive. Weyl was also a very influential figure in all three major fields of 20th-century physics: relativity, unified field theory and quantum mechanics. Because of his contributions to Schrödinger, many think the latter's famous result should be named Schrödinger-Weyl Wave Equation.

Vladimir Vizgin wrote "To this day, Weyl's [unified field] theory astounds all in the depth of its ideas, its mathematical simplicity, and the elegance of its realization." Weyl once wrote: "My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful."

Felix Klein
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."
George Pólya
George Pólya (Pólya György) did significant work in several fields: complex analysis, probability, geometry, algebraic number theory, and combinatorics, but is most noted for his teaching How to Solve It, the craft of problem posing and proof. He is also famous for the Pólya Enumeration Theorem. Several other important theorems he proved include the Pólya-Vinogradov Inequality of number theory, the Pólya-Szego Inequality of functional analysis, and the Pólya Inequality of measure theory. He introduced the Hilbert-Pólya Conjecture that the Riemann Hypothesis might be a consequence of spectral theory; he introduced the famous "All horses are the same color" example of inductive fallacy; he named the Central Limit Theorem of statistics. Pólya was the "teacher par excellence": he wrote top books on multiple subjects; his successful students included John von Neumann. His work on plane symmetry groups directly inspired Escher's drawings. Having huge breadth and influence, Pólya has been called "the most influential mathematician of the 20th century."
Godfrey H. Hardy
Hardy was an extremely prolific research mathematician who did important work in analysis (especially the theory of integration), number theory, global analysis, and analytic number theory. He proved several important theorems about numbers, for example that Riemann's zeta function has infinitely many zeros with real part 1/2. He was also an excellent teacher and wrote several excellent textbooks, as well as a famous treatise on the mathematical mind. He abhorred applied mathematics, treating mathematics as a creative art; yet his work has found application in population genetics, cryptography, thermodynamics and particle physics.
John Wallis
Wallis began his life as a savant at arithmetic (it is said he once calculated the square root of a 53-digit number to help him sleep and remembered the result in the morning), a medical student (he may have contributed to the concept of blood circulation), and theologian, but went on to become perhaps the most brilliant and influential English mathematician before Newton. He made major advances in analytic geometry, but also contributions to algebra, geometry and trigonometry. Unlike his contemporary, Huygens who took inspiration from Euclid's rigorous geometry, Wallis embraced the new analytic methods of Descartes and Fermat. He is especially famous for using negative and fractional exponents (though Oresme had introduced fractional exponents three centuries earlier), taking the areas of curves, and treating inelastic collisions (he and Huygens were first to develop the law of momentum conservation). He was the first European to solve Pell's Equation. Like Vieta, Wallis was a code-breaker, helping the Commonwealth side (though he later petitioned against the beheading of King Charles I). He was the first great mathematician to consider complex numbers legitimate; and first to use the symbol ∞. Wallis coined several terms including continued fraction, induction, interpolation, mantissa, and hypergeometric series.
Girolamo Cardano
Girolamo Cardano (or Jerome Cardan) was a highly respected physician and was first to describe typhoid fever. He was also an accomplished gambler and chess player and wrote an early book on probability. He was also a remarkable inventor: the combination lock, an advanced gimbal, a ciphering tool, and the Cardan shaft with universal joints are all his inventions and are in use to this day. (The U-joint is sometimes called the Cardan joint.) He also helped develop the camera obscura. Cardano made contributions to physics: he noted that projectile trajectories are parabolas, and may have been first to note the impossibility of perpetual motion machines. He did work in philosophy, geology, hydrodynamics, music; he wrote books on medicine and an encyclopedia of natural science.
Christiaan Huygens
Christiaan Huygens (or Hugens, Huyghens) was second only to Newton as the greatest mechanist of his era. Although an excellent mathematician, he is much more famous for his physical theories and inventions. He developed laws of motion before Newton, including the inverse-square law of gravitation, centripetal force, and treatment of solid bodies rather than point approximations; he (and Wallis) were first to state the law of momentum conservation correctly. He advanced the wave ("undulatory") theory of light, a key concept being Huygen's Principle, that each point on a wave front acts as a new source of radiation. His optical discoveries include explanations for polarization and phenomena like haloes. (Because of Newton's high reputation and corpuscular theory of light, Huygens' superior wave theory was largely ignored until the 19th-century work of Young, Fresnel, and Maxwell. Later, Planck, Einstein and Bohr, partly anticipated by Hamilton, developed the modern notion of wave-particle duality.)
Bernhard Riemann
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 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.
Alfred Tarski
Alfred Tarski (born Alfred Tajtelbaum) was one of the greatest and most prolific logicians ever, but also made advances in set theory, measure theory, topology, algebra, group theory, computability theory, metamathematics, and geometry. He was also acclaimed as a teacher. Although he achieved fame at an early age with the Banach-Tarski Paradox, his greatest achievements were in formal logic. He wrote on the definition of truth, developed model theory, and investigated the completeness questions which also intrigued Gödel. He proved several important systems to be incomplete, but also established completeness results for real arithmetic and geometry. His most famous result may be Tarski's Undefinability Theorem, which is related to Gödel's Incompleteness Theorem but more powerful. Several other theorems, theories and paradoxes are named after Tarski including Tarski-Grothendieck Set Theory, Tarski's Fixed-Point Theorem of lattice theory (from which the famous Cantor-Bernstein Theorem is a simple corollary), and a new derivation of the Axiom of Choice (which Lebesgue refused to publish because "an implication between two false propositions is of no interest"). Tarski was first to enunciate the remarkable fact that the Generalized Continuum Hypothesis implies the Axiom of Choice, although proof had to wait for Sierpinski. Tarski's other notable accomplishments include his cylindrical algebra, ordinal algebra, universal algebra, and an elegant and novel axiomatic basis of geometry.
Pierre-Simon Laplace
Laplace was the preeminent mathematical astronomer, and is often called the "French Newton." His masterpiece was Mecanique Celeste which redeveloped and improved Newton's work on planetary motions using calculus. While Newton had shown that the two-body gravitation problem led to orbits which were ellipses (or other conic sections), Laplace was more interested in the much more difficult problems involving three or more bodies. (Would Jupiter's pull on Saturn eventually propel Saturn into a closer orbit, or was Saturn's orbit stable for eternity?) Laplace's equations had the optimistic outcome that the solar system was stable.
Jean-Pierre Serre
Serre did important work with spectral sequences and algebraic methods, revolutionizing the study of algebraic topology and algebraic geometry, especially homotopy groups and sheaves. Hermann Weyl praised Serre's work strongly, saying it gave an important new algebraic basis to analysis. He collaborated with Grothendieck and Pierre Deligne, helped resolve the Weil conjectures, and contributed indirectly to the recent proof of Fermat's Last Theorem. His wide range of research areas also includes number theory, bundles, fibrations, p-adic modular forms, Galois representation theory, and more. Serre has been much honored: he is the youngest ever to win a Fields Medal; 49 years after his Fields Medal he became the first recipient of the Abel Prize.
Srinivasa Ramanujan
Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. While some of these were old theorems or just curiosities, many were brilliant new theorems with very difficult proofs. For example, he found a beautiful identity connecting Poisson summation to the Möbius function. Ramanujan might be almost unknown today, except that his letter caught the eye of Godfrey Hardy, who saw remarkable, almost inexplicable formulae which "must be true, because if they were not true, no one would have had the imagination to invent them." Ramanujan's specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, gamma functions, "mock theta" functions, hypergeometric series, and "highly composite" numbers. Ramanujan's "Master Theorem" has wide application in analysis, and has been applied to the evaluation of Feynman diagrams. Much of his best work was done in collaboration with Hardy, for example a proof that almost all numbers n have about log log n prime factors (a result which developed into probabilistic number theory). Much of his methodology, including unusual ideas about divergent series, was his own invention. (As a young man he made the absurd claim that 1+2+3+4+... = -1/12. Later it was noticed that this claim translates to a true statement about the Riemann zeta function, with which Ramanujan was unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi.
Émile Borel
Borel exhibited great talent while still in his teens, soon practically founded modern measure theory, and received several honors and prizes. Among his famous theorems is the Heine-Borel Covering Theorem. He also did important work in several other fields of mathematics, including divergent series, quasi-analytic functions, differential equations, number theory, complex analysis, theory of functions, geometry, probability theory, and game theory. Relating measure theory to probabilities, he introduced concepts like normal numbers and the Borel-Kolmogorov paradox. He also did work in relativity and the philosophy of science. He anticipated the concept of chaos, inspiring Poincaré. Borel combined great creativity with strong analytic power; however he was especially interested in applications, philosophy, and education, so didn't pursue the tedium of rigorous development and proof; for this reason his great importance as a theorist is often underestimated. Borel was decorated for valor in World War I, entered politics between the Wars, and joined the French Resistance during World War II.
Michael Atiyah
Atiyah's career has had extraordinary breadth and depth. He advanced the theory of vector bundles; this developed into topological K-theory and the Atiyah-Singer Index Theorem. This Index Theorem is considered one of the most far-reaching theorems ever, subsuming famous old results (Descartes' total angular defect, Euler's topological characteristic), important 19th-century theorems (Gauss-Bonnet, Riemann-Roch), and incorporating important work by Weil and especially Shiing-Shen Chern. It is a key to the study of high-dimension spaces, differential geometry, and equation solving. Several other key results are named after Atiyah, e.g. the Atiyah-Bott Fixed-Point Theorem, the Atiyah-Segal Completion Theorem, and the Atiyah-Hirzebruch spectral sequence. Atiyah's work developed important connections not only between topology and analysis, but with modern physics; Atiyah himself has been a key figure in the development of string theory. This work, and Atiyah-inspired work in gauge theory, restored a close relationship between leading edge research in mathematics and physics. Atiyah is known as a vivacious genius in person, inspiring many, e.g. Edward Witten. Along with Serre, Atiyah is often considered to be one of the very greatest living mathematicians.
Atiyah once said a mathematician must sometimes "freely float in the atmosphere like a poet and imagine the whole universe of possibilities, and hope that eventually you come down to Earth somewhere else."
Shiing-Shen Chern
Shiing-Shen Chern (Chen Xingshen) studied under Élie Cartan, and became perhaps the greatest master of differential geometry. He is especially noted for his work in algebraic geometry, topology and fiber bundles, developing his Chern characters (in a paper with "a tremendous number of geometrical jewels"), developing Chern-Weil theory, the Chern-Simons invariants, and especially for his brilliant generalization of the Gauss-Bonnet Theorem to multiple dimensions. His work had a major influence in several fields of modern mathematics as well as gauge theories of physics. Chern was an important influence in China and a highly renowned and successful teacher: one of his students (Yau) won the Fields Medal, another (Yang) the Nobel Prize in physics. Chern himself was the first Asian to win the prestigious Wolf Prize.
Alexandre Grothendieck
Grothendieck has done brilliant work in several areas of mathematics including number theory, geometry, topology, and functional analysis, but especially in the fields of algebraic geometry and category theory, both of which he revolutionized. He is especially noted for his invention of the Theory of Schemes, and other methods to unify different branches of mathematics. He applied algebraic geometry to number theory; applied methods of topology to set theory; etc. Grothendieck is considered a master of abstraction, rigor and presentation. He has produced many important and deep results in homological algebra, most notably his etale cohomology. With these new methods, Grothendieck and his outstanding student Pierre Deligne were able to prove the Weil Conjectures. Grothendieck also developed the theory of sheafs, generalized the Riemann-Roch Theorem to revolutionize K-theory, developed Grothendieck categories, crystalline cohomology, infinity-stacks and more. The guiding principle behind much of Grothendieck's work has been Topos Theory, which he invented to harness the methods of topology. These methods and results have redirected several diverse branches of modern mathematics including number theory, algebraic topology, and representation theory. Among Grothendieck's famous results was his Fundamental Theorem in the Metric Theory of Tensor Products, which was inspired by Littlewood's proof of the 4/3 Inequality.
Grothendieck's radical religious and political philosophies led him to retire from public life while still in his prime, but he is widely regarded as the greatest mathematician of the 20th century, and indeed one of the greatest geniuses ever.
Andrey Kolmogorov
Kolmogorov had a powerful intellect and excelled in many fields. As a youth he dazzled his teachers by constructing toys that appeared to be "Perpetual Motion Machines." At the age of 19, he achieved fame by finding a Fourier series that diverges almost everywhere, and decided to devote himself to mathematics. He is considered the founder of the fields of intuitionistic logic, algorithmic complexity theory, and (by applying measure theory) modern probability theory. He also excelled in topology, set theory, trigonometric series, and random processes. He and his student Vladimir Arnold proved the surprising Superposition Theorem, which not only solved Hilbert's 13th Problem, but went far beyond it. He and Arnold also developed the "magnificent" KAM Theorem, which quantifies how strong a perturbation must be to upset a quasiperiodic dynamical system. Kolmogorov's axioms of probability are considered a partial solution of Hilbert's 6th Problem. He made important contributions to the constructivist ideas of Kronecker and Brouwer. While Kolmogorov's work in probability theory had direct applications to physics, Kolmogorov also did work in physics directly, especially the study of turbulence. There are dozens of notions named after Kolmogorov, such as the Kolmogorov Backward Equation, the Borel-Kolmogorov Paradox, and the intriguing Zero-One Law of "tail events" among random variables.
André Weil
Weil made profound contributions to several areas of mathematics, especially algebraic geometry, which he showed to have deep connections with number theory. His Weil conjectures were very influential; these and other works laid the groundwork for some of Grothendieck's work. Weil proved a special case of the Riemann Hypothesis; he contributed, at least indirectly, to the recent proof of Fermat's Last Theorem; he also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. He invented several new concepts including vector bundles, and uniform space. His work has found applications in particle physics and string theory. He is considered to be one of the most influential of modern mathematicians.
Weil's biography is interesting. He studied Sanskrit as a child, loved to travel, taught at a Muslim university in India for two years (intending to teach French civilization), wrote as a young man under the famous pseudonym Nicolas Bourbaki, spent time in prison during World War II as a Jewish objector, was almost executed as a spy, escaped to America, and eventually joined Princeton's Institute for Advanced Studies. He once wrote: "Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thought succeeds another as if miraculously."
Augustin Cauchy
Cauchy was extraordinarily prodigious, prolific and inventive. Home-schooled, he awed famous mathematicians at an early age. In contrast to Gauss and Newton, he was almost over-eager to publish; in his day his fame surpassed that of Gauss and has continued to grow. Cauchy did significant work in analysis, algebra, number theory and discrete topology. His most important contributions included convergence criteria for infinite series, the "theory of substitutions" (permutation group theory), and especially his insistence on rigorous proofs.
Élie Cartan
Cartan worked in the theory of Lie groups and Lie algebras, applying methods of topology, geometry and invariant theory to Lie theory, and classifying all Lie groups. This work was so significant that Cartan, rather than Lie, is considered the most important developer of the theory of Lie groups. Using Lie theory and ideas like his Method of Prolongation he advanced the theories of differential equations and differential geometry. Cartan introduced several new concepts including algebraic group, exterior differential forms, spinors, moving frames, Cartan connections. He proved several important theorems, e.g. Schläfli's Conjecture about embedding Riemann metrics, and fundamental theorems about symmetric Riemann spaces. He made a key contribution to Einstein's general relativity, based on what is now called Riemann-Cartan geometry. Cartan's methods were so original as to be fully appreciated only recently; many now consider him to be one of the greatest mathematicians of his era. In 1938 Weyl called him "the greatest living master in differential geometry."
Apollonius of Perga
Apollonius Pergaeus, called "The Great Geometer," is sometimes considered the second greatest of ancient Greek mathematicians. (Euclid, Eudoxus and Archytas are other candidates for this honor.) His writings on conic sections have been studied until modern times; he invented the names for parabola, hyperbola and ellipse; he developed methods for normals and curvature. Although astronomers eventually concluded it was not physically correct, Apollonius developed the "epicycle and deferent" model of planetary orbits, and proved important theorems in this area. He deliberately emphasized the beauty of pure, rather than applied, mathematics, saying his theorems were "worthy of acceptance for the sake of the demonstrations themselves." The following generalization of the Pythagorean Theorem, where M is the midpoint of BC, is called Apollonius' Theorem: AB 2 + AC 2 = 2(AM 2 + BM 2).
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, discovered the Laplace transform before Laplace did, and developed terminology and notation (e.g. the use of f'(x) and f''(x) for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's Conjecture (p divides (p-1)! + 1 when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and that n·x2 + 1 = y2 has solutions for every positive non-square integer n.
Jacques Hadamard
Hadamard made revolutionary advances in several different areas of mathematics, especially complex analysis, analytic number theory, differential geometry, partial differential equations, symbolic dynamics, chaos theory, matrix theory, and Markov chains; for this reason he is sometimes called the "Last Universal Mathematician." He also made contributions to physics. One of the most famous results in mathematics is the Prime Number Theorem, that there are approximately n/log n primes less than n. This result was conjectured by Legendre and Gauss, attacked cleverly by Riemann and Chebyshev, and finally, by building on Riemann's work, proved by Hadamard and Vallee-Poussin. (Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.) Several other important theorems are named after Hadamard (e.g. his Inequality of Determinants), and some of his theorems are named after others (Hadamard was first to prove Brouwer's Fixed-Point Theorem for arbitrarily many dimensions). Hadamard was also influential in promoting others' work: He is noted for his survey of Poincaré's work; his staunch defense of the Axiom of Choice led to the acceptance of Zermelo's work. Hadamard was a successful teacher, with André Weil, Maurice Fréchet, and others acknowledging him as key inspiration. Like many great mathematicians he emphasized the importance of intuition, writing "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it."
Georg Cantor
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.
Carl Ludwig Siegel
Carl Siegel became famous when his doctoral dissertation established a key result in Diophantine approximations. He continued with contributions to several branches of analytic and algebraic number theory, including arithmetic geometry and quadratic forms. He also did seminal work with Riemann's zeta function, Dedekind's zeta functions, transcendental number theory, discontinuous groups, the 3-body problem in celestial mechanics, and symplectic geometry. In complex analysis he developed Siegel modular forms, which have wide application in math and physics. He may share credit with Alexander Gelfond for the solution to Hilbert's 7th Problem, Siegel admired the "simplicity and honesty" of masters like Gauss, Lagrange and Hardy and lamented the modern "trend for senseless abstraction." He and Israel Gelfand were the first two winners of the Wolf Prize in Mathematics. Atle Selberg called him a "devastatingly impressive" mathematician who did things that "seemed impossible." André Weil declared that Siegel was the greatest mathematician of the first half of the 20th century.
Sophus Lie
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."
Emmy Noether
Noether was an innovative researcher who was considered the greatest master of abstract algebra ever; her advances included a new theory of ideals, the inverse Galois problem, and the general theory of commutative rings. She originated novel reasoning methods, especially one based on "chain conditions," which advanced invariant theory and abstract algebra; her insistence on generalization led to a unified theory of modules and Noetherian rings. Her approaches tended to unify disparate areas (algebra, geometry, topology, logic) and led eventually to modern category theory. Her invention of Betti homology groups led to algebraic topology, and thus revolutionized topology.
Adrien-Marie Legendre
Legendre was an outstanding mathematician who did important work in plane and solid geometry, spherical trigonometry, celestial mechanics and other areas of physics, and especially elliptic integrals and number theory. He found key results in the theories of sums of squares and sums of k-gonal numbers. He also made key contributions in several areas of analysis: he invented the Legendre transform and Legendre polynomials; the notation for partial derivatives is due to him. He invented the Legendre symbol; invented the study of zonal harmonics; proved that π2 was irrational (the irrationality of π had already been proved by Lambert); and wrote important textbooks in several fields. Although he never accepted non-Euclidean geometry, and had spent much time trying to prove the Parallel Postulate, his inspiring geometry text remained a standard until the 20th century. As one of France's premier mathematicians, Legendre did other significant work, promoting the careers of Lagrange and Laplace, developing trig tables, geodesic projects, etc.
Richard Dedekind
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."
Leonardo Bonacci 'Fibonacci'
Leonardo (known today as Fibonacci) introduced the decimal system and other new methods of arithmetic to Europe, and relayed the mathematics of the Hindus, Persians, and Arabs. Others had translated Islamic mathematics, e.g. the works of al-Khowârizmi, into Latin, but Leonardo was the influential teacher. He also re-introduced older Greek ideas like Mersenne numbers and Diophantine equations. Leonardo's writings cover a very broad range including new theorems of geometry, methods to construct and convert Egyptian fractions (which were still in wide use), irrational numbers, the Chinese Remainder Theorem, theorems about Pythagorean triplets, and the series 1, 1, 2, 3, 5, 8, 13, .... which is now linked with the name Fibonacci. In addition to his great historic importance and fame (he was a favorite of Emperor Frederick II), Leonardo `Fibonacci' is called "the greatest number theorist between Diophantus and Fermat" and "the most talented mathematician of the Middle Ages."
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