Earliest mathematicians
Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19) in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where 4500-year old clay tablets show multiplication and division problems; the first abacus may be about this old. By 3600 years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms, using a primitive place-value system (in base 60, not 10). Babylonians were familiar with the Pythagorean theorem, quadratic equations, even cubic equations (though they didn't have a general solution for these), and eventually even developed methods to estimate terms for compound interest.
Also at least 3600 years ago, the Egyptian scribe Ahmes produced a famous manuscript (now called the Rhind Papyrus), itself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions. (Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians. To divide 17 grain bushels among 21 workers, the equation 17/21 = 1/2 + 1/6 + 1/7 has practical value, especially when compared with the "greedy" decomposition 17/21 = 1/2 + 1/4 + 1/17 + 1/1428.)
While Egyptians may have had more advanced geometry, Babylon was much more advanced at arithmetic and algebra. This was probably due, at least in part, to their place-value system. But although their base-60 system survives (e.g. in the division of hours and degrees into minutes and seconds) the Babylonian notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII to denote 417+43/60, was unwieldy compared to the "ten digits of the Hindus."
The Egyptians used the approximation π ≈ (4/3)4 (derived from the idea that a circle of diameter 9 has about the same area as a square of side 8). Although the ancient Hindu mathematician Apastambha had achieved a good approximation for √2, and the ancient Babylonians an ever better √2, neither of these ancient cultures achieved a π approximation as good as Egypt's, or better than π ≈ 25/8, until the Alexandrian era.
Early Vedic mathematicians
The greatest mathematics before the Golden Age of Greece was in India's early Vedic (Hindu) civilization. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals. The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about 1300 BC and used geometry and elementary trigonometry for his astronomy. Baudhayana lived about 800 BC and also wrote on algebra and geometry; Yajnavalkya lived about the same time and is credited with the then-best approximation to π. Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations.
Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills. Ancient China certainly developed mathematics, though little written evidence survives prior to Chang Tshang's famous book.
Pythagoras of Samos (ca 578-505 BC) Greek domain
Pythagoras, who is sometimes called the "First Philosopher," studied under Anaximander, Egyptians, Babylonians, and the mystic Pherekydes (from whom Pythagoras acquired a belief in reincarnation); he became the most influential of early Greek mathematicians. He is credited with being first to use axioms and deductive proofs, so his influence on Plato and Euclid may be enormous. He and his students (the "Pythagoreans") were ascetic mystics for whom mathematics was partly a spiritual tool. (Some occultists treat Pythagoras as a wizard and founding mystic philosopher.) Pythagoras was very interested in astronomy and recognized that the Earth was a globe similar to the other planets. He believed thinking was located in the brain rather than heart. The words "philosophy" and "mathematics" are said to have been coined by Pythagoras. Despite Pythagoras' historical importance I may have ranked him too high: many results of the Pythagoreans were due to his students; none of their writings survive; and what is known is reported second-hand, and possibly exaggerated, by Plato and others. His students included Hippasus of Metapontum, perhaps the famous physician Alcmaeon, Milo of Croton, and Croton's daughter Theano (who may have been Pythagoras's wife). The term "Pythagorean" was also adopted by many disciples who lived later; these disciples include Philolaus of Croton, the natural philosopher Empedocles, and several other famous Greeks. Pythagoras' successor was apparently Theano herself: the Pythagoreans were one of the few ancient schools to practice gender equality.
Pythagoras discovered that harmonious intervals in music are based on simple rational numbers. This led to a fascination with integers and mystic numerology; he is sometimes called the "Father of Numbers" and once said "Number rules the universe." (About the mathematical basis of music, Leibniz later wrote, "Music is the pleasure the human soul experiences from counting without being aware that it is counting." Other mathematicians who investigated the arithmetic of music included Huygens, Euler and Simon Stevin.)
The Pythagorean Theorem was known long before Pythagoras, but he is often credited with the first proof. (Apastambha proved it in India at about the same time, and some theorize that Pythagoras journeyed to India and learned of the proof there.) He also discovered the simple parametric form of Pythagorean triplets (xx-yy, 2xy, xx+yy). Other discoveries of the Pythagorean school include the concepts of perfect and amicable numbers, polygonal numbers, golden ratio (attributed to Theano), the five regular solids (attributed to Pythagoras himself), and irrational numbers (attributed to Hippasus). It is said that the discovery of irrational numbers upset the Pythagoreans so much they tossed Hippasus into the ocean! (Another version has Hippasus banished for revealing the secret for constructing the sphere which circumscribes a dodecahedron.)
The famous successors of Thales and Pythagoras included Parmenides of Elea (ca 515-440 BC), Zeno of Elea (see below), Hippocrates of Chios (see below), Plato of Athens (ca 428-348 BC), Theaetetus (ca 414-369 BC), and Archytas (see below). These early Greeks ushered in a Golden Age of Mathematics and Philosophy unequaled in Europe until the Renaissance. The emphasis was on pure, rather than practical, mathematics. Plato (who ranks #40 on Michael Hart's famous list of the Most Influential Persons in History) decreed that his scholars should do geometric construction solely with compass and straight-edge rather than with "carpenter's tools" like rulers and protractors.
Panini (of Shalatula) (ca 520-460 BC) Gandhara (India)
Panini's great accomplishment was his study of the Sanskrit language, especially in his text Ashtadhyayi. Although this work might be considered the very first study of linguistics or grammar, it used a non-obvious elegance that would not be equalled in the West until the 20th century. Linguistics may seem an unlikely qualification for a "great mathematician," but language theory is a field of mathematics. The works of eminent 20th-century linguists and computer scientists like Chomsky, Backus, Post and Church are seen to resemble Panini's work 24 centuries earlier. Panini's systematic study of Sanskrit may have inspired the development of Indian science and algebra. Panini has been called "the Indian Euclid" since the rigor of his grammar is comparable to Euclid's geometry. Although his great texts have been preserved, little else is known about Panini. Some scholars would place his dates a century later than shown here; he may or may not have been the same person as the famous poet Panini. In any case, he was the very last Vedic Sanskrit scholar by definition: his text formed the transition to the Classic Sanskrit period. Panini has been called "one of the most innovative people in the whole development of knowledge." Zeno of Elea (ca 495-435 BC) Greek domain
Zeno, a student of Parmenides, had great fame in ancient Greece. This fame, which continues to the present-day, is largely due to his paradoxes of infinities and infinitesimals, e.g. his argument that Achilles can never catch the tortoise (whenever Achilles arrives at the tortoise's last position, the tortoise has moved on). Although some regard these paradoxes as simple fallacies, they have been contemplated for many centuries. It is due to these paradoxes that the use of infinitesimals, which provides the basis for mathematical analysis, has been regarded as a non-rigorous heuristic and is finally viewed as sound only after the work of the great 19th-century rigorists, Dedekind and Weierstrass.
Hippocrates of Chios (ca 470-410 BC) Greek domain
Hippocrates (no relation to the famous physician) wrote his own Elements more than a century before Euclid. Only fragments survive but it apparently used axiomatic-based proofs similar to Euclid's and contains many of the same theorems. Hippocrates is said to have invented the reductio ad absurdem proof method. Hippocrates is most famous for his work on the three ancient geometric quandaries: his work on cube-doubling (the Delian Problem) laid the groundwork for successful efforts by Archytas and others; his circle quadrature was of course ultimately unsuccessful but he did prove ingenious theorems about "lunes" (certain circle fragments); and some claim Hippocrates was first to trisect the general angle. (Doubling the cube and angle trisection are often called "impossible," but they are impossible only when restricted to collapsing compass and unmarkable straightedge. There are ingenious solutions available with other tools.) Hippocrates also did work in algebra and rudimentary analysis.
Archytas of Tarentum (ca 420-350 BC) Greek domain
Archytas was an important statesman as well as philosopher. He studied under Philolaus of Croton, was a friend of Plato, and tutored Eudoxus and Menaechmus. In addition to discoveries always attributed to him, he may be the source of several of Euclid's theorems, and some works attributed to Eudoxus and perhaps Pythagoras. Recently it has been shown that the magnificent Mechanical Problems attributed to (pseudo-)Aristotle were probably actually written by Archytas, making him one of the greatest mathematicians of antiquity. Archytas introduced "motion" to geometry, rotating curves to produce solids. If his writings had survived he'd surely be considered one of the most brilliant and innovative geometers of antiquity. Archytas' most famous mathematical achievement was "doubling the cube" (constructing a line segment larger than another by the factor cube-root of two). Although others solved the problem with other techniques, Archytas' solution for cube doubling was astounding because it wasn't achieved in the plane, but involved the intersection of three-dimensional bodies. This construction (which introduced the "Archytas Curve") has been called "a tour de force of the spatial imagination." He invented the term "harmonic mean" and worked with geometric means as well (proving that consecutive integers never have rational geometric mean). He was a true polymath: he advanced the theory of music far beyond Pythagoras; studied sound, optics and cosmology; invented the pulley (and a rattle to occupy infants!); wrote about the lever; developed the curriculum called quadrivium; and is supposed to have built a steam-powered wooden bird which flew for 200 meters. Archytas is sometimes called the Father of Mathematical Mechanics.
John Horton Conway (1937-) Britain
Conway has done pioneering work in a very broad range of mathematics including knot theory, number theory, group theory, lattice theory, combinatorial game theory, geometry, quaternions, tilings, and cellular automaton theory. He started his career by proving a case of Waring's conjecture, but achieved fame when he discovered the largest then-known sporadic group (the symmetry group of the Leech lattice); this sporadic group is now known to be second in size only to the "Monster Group," with which Conway also worked. Conway's fertile creativity has produced a cornucopia of fascinating inventions: markable straight-edge construction of the regular heptagon (a feat also achieved by Archimedes), a nowhere-continuous function that has the Intermediate Value property, the Conway box function, the aperiodic pinwheel tiling, a representation of symmetric polyhedra, his chained-arrow notation for large numbers, and many results and conjectures in recreational mathematics. He found the simplest proof for Morley's Trisector Theorem (sometimes called the best result in simple plane geometry since ancient Greece). He proved an unusual theorem about quantum physics: "If experimenters have free will, then so do elementary particles." His most famous construction is the computationally complete automaton known as the Game of Life. His most important theoretical invention, however, may be his surreal numbers incorporating infinitesimals; he invented them to solve combinatorial games like Go, but they have pure mathematical significance as the largest possible ordered field. Although Conway hasn't won the highest mathematics prizes, nor perhaps proved deep theoretical theorems, his great creativity and breadth qualifies him as one of the greatest living mathematicians. Srinivasa Ramanujan Iyengar (1887-1920) India
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. He 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. 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 inspired 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. Ramanujan's most famous work was with the partition enumeration function p(), Hardy guessing that some of these discoveries would have been delayed at least a century without Ramanujan. Together, Hardy and Ramanujan developed an analytic approximation to p(). (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)
Many of Ramanujan's other results would also probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.) Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.) Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused on real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever.
Because of its fast convergence, an odd-looking formula of Ramanujan is often used to calculate π:
992 / π = √8 ∑k=0,∞ (4k! (1103+26390 k) / (k!4 3964k))
Brahmagupta `Bhillamalacarya' (589-668) Rajasthan (India)
No one person gets unique credit for the invention of the decimal system but Brahmagupta's textbook Brahmasphutasiddhanta was very influential, and is sometimes considered the first textbook "to treat zero as a number in its own right." It also treated negative numbers. (Others claim these were first seen 800 years earlier in Chang Tshang's Chinese text and were implicit in what survives of earlier Hindu works, but Brahmagupta's text discussed them lucidly.)
Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala') made great advances in arithmetic, algebra, numeric analysis, and geometry. Several theorems bear his name, including the formula for the area of a cyclic quadrilateral:
16 A2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)
Another famous Brahmagupta theorem dealing with such quadrilaterals can be phrased "In a circle, if the chords AB and CD are perpendicular and intersect at E, then the line from E which bisects AC will be perpendicular to BD." Proving Brahmagupta's theorems are good challenges even today.
In addition to his famous writings on practical mathematics and his ingenious theorems of geometry, Brahmagupta solved the general quadratic equation, and worked on Diophantine and Pell's equations. He proved the Brahmagupta-Fibonacci Identity (the set of sums of two squares is closed under multiplication). He applied mathematics to astronomy, predicting eclipses, etc. Muhammed `Abu Jafar' ibn Musâ al-Khowârizmi (ca 780-850) Persia, Iraq
Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian who worked as a mathematician, astronomer and geographer early in the Golden Age of Islamic science. He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi presented general methods. The word algorithm is borrowed from Al-Khowârizmi's name. There were several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, but Al-Khowârizmi was one of the earliest and most influential.
Abu Yusuf Ya'qub ibn Ishaq al-Kindi (803-873) Iraq
Al-Kindi (called Alkindus in the West) wrote on diverse philosophical subjects, physics, optics, astronomy, music, psychology, medicine, chemistry, and more. He invented pharmaceutical methods, perfumes, and distilling of alcohol. In mathematics, he popularized the use of the decimal system, developed spherical geometry, wrote on many other topics and was a pioneer of cryptography (code-breaking). (Al-Kindi, called The Arab Philosopher, can not be considered among the greatest of mathematicians, but was one of the most influential general scientists between Aristotle and da Vinci.)
Al-Sabi Thabit ibn Qurra al-Harrani (836-901) Harran, Iraq
Thabit produced important books in philosophy (including perhaps the famous mystic work De Imaginibus), mechanics, astronomy, and especially several mathematical fields: analysis, non-Euclidean geometry, trigonometry, arithmetic, number theory. He developed an important new cosmology superior to Ptolemy's (and which, though it was not heliocentric, may have inspired Copernicus). He was perhaps the first great mathematician to take the important step of emphasizing real numbers rather than either rational numbers or geometric sizes. He worked in plane and spherical trigonometry, in cubic equations. He was an earlier practitioner of calculus and seems to have been first to take the integral of √x. He also may have been first to calculate the area of an ellipse, and first to calculate the volume of a paraboloid. He also worked in number theory where he is especially famous for his theorem about amicable numbers. While many of his discoveries in geometry, plane and spherical trigonometry, and analysis (parabola quadrature, trigonometric law, principle of lever) duplicated work by Archimedes and Pappus, Thabit's list of novel achievements is still impressive. Among the several great and famous Baghdad geometers, Thabit may have had the greatest genius. By: Tharshini Jayaseelan
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