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Gherverghese On Non-European Mathematics

a deep-rooted historiographical bias in the selection of interpretation of facts [wherein] mathematical activity outside Europe has as a consequence been ignored, devalued or distorted (Crest Of The Peacock, First Oxford Edition, 2000, p. 3).

The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks.

[Mathematics] finally secured a new grip on life in the highly congenial soil of Greece and waxed strongly for a short period...With the decline of Greek civilization the plant remained dormant for a thousand years...when the plant was transported to Europe proper and once more imbedded in fertile soil.

ignores a considerable body of research evidence pointing to development of mathematics in Mesopotamia, Egypt, China, pre-Columbian America, India, and the Arab world that had come to light [by time Kline wrote his book].

My Decisive Refutation

Greek indebtedness to Egyptians and Phoenicians

Knowledge of the science of numbers possessed by the Phoenicians

Knowledge of the science of numbers possessed by the Egyptians

Knowledge of the science of geometry possessed by the Egyptians

From this rapid sketch it will be seen that the work of the Arabs (including therein writers who wrote in Arabia and lived under Eastern Mohammedan rule) in arithmetic, algebra, and trigonometry was of a high order of excellence (p. 135).

The object of this book is to advance the thesis that mathematics has been a major cultural force in Western civilization (1978 ed., p. vii).

Egypt developed a worthy type of mathematics before 1000 BC and then stagnated, Babylonia did the same, and China followed a similar course (p. 96).

In Asia the gloom was particularly oppressive in the 16th century. India was intellectually dead [...] The introduction of Western civilization into India, China...is interesting because of its diverse effects. As to India, mathematics was already stagnant, and the European influence gave it no stimulus...China, which had done so much in algebra, was content in the 17th century to adopt European astronomy while allowing her own undoubted abilities to lie dormant [...] Except for the case of Japan, the Orient lost its initiative in mathematics...India produced nothing that was distinctive in the 18th or 19th century. China, while occasionally protesting against Western mathematics, in reality sacrificed on the altar of the Jesuit missionaries her own originality in the science (pp. 350, 435, 533).

List Of Continuous European Mathematical Creativity

1323-1382 Nicole Oresme, French: System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series

1446-1517 Luca Pacioli, Italian: Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus

1499-1557 Niccolò Fontana Tartaglia, Italian: Formula for solving all types of cubic equations, involving first real use of complex numbers (combinations of real and imaginary numbers), Tartaglia's Triangle (earlier version of Pascal's Triangle)

1501-1576 Gerolamo Cardano, Italian: Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)

1522-1565 Lodovico Ferrari, Italian: Devised formula for solution of quartic equations

1550-1617 John Napier, British: Invention of natural logarithms, popularized the use of the decimal point, Napier's Bones tool for lattice multiplication

1588-1648 Marin Mersenne, French: Clearing house for mathematical thought during 17th Century, Mersenne primes (prime numbers that are one less than a power of 2)

1591-1661 Girard Desargues, French: Early development of projective geometry and "point at infinity," perspective theorem

1596-1650 René Descartes, French: Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents

1598-1647 Bonaventura Cavalieri, Italian: "Method of indivisibles” paved way for the later development of infinitesimal calculus

1601-1665 Pierre de Fermat, French: Discovered many new numbers patterns and theorems (including Little Theorem, Two-Square Thereom and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory

1616-1703 John Wallis, British: Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers

1623-1662 Blaise Pascal, French: Pioneer (with Fermat) of probability theory, Pascal's Triangle of binomial coefficients

1643-1727 Isaac Newton, British: Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series

1646-1716 Gottfried Leibniz, German: Independently developed infinitesimal calculus (his calculus notation is still used), also practical calculating machine using binary system (forerunner of the computer), solved linear equations using a matrix

1654-1705 Jacob Bernoulli, Swiss: Helped to consolidate infinitesimal calculus, developed a technique for solving separable differential equations, added a theory of permutations and combinations to probability theory, Bernoulli Numbers sequence, transcendental curves

1667-1748 Johann Bernoulli, Swiss: Further developed infinitesimal calculus, including the "calculus of variation," functions for curve of fastest descent (brachistochrone) and catenary curve

1667-1754 Abraham de Moivre, French: De Moivre's formula, development of analytic geometry, first statement of the formula for the normal distribution curve, probability theory

1690-1764 Christian Goldbach, German: Goldbach Conjecture, Goldbach-Euler Theorem on perfect powers

1707-1783 Leonhard Euler, Swiss: Made important contributions in almost all fields and found unexpected links between different fields, proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks

1728-1777 Johann Lambert, Swiss: Rigorous proof that π is irrational, introduced hyperbolic functions into trigonometry, made conjectures on non-Euclidean space and hyperbolic triangles

1736-1813 Joseph Louis Lagrange, Italian/French: Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange's theorem of finite groups, four-square theorem, mean value theorem

1746-1818 Gaspard Monge, French: Inventor of descriptive geometry, orthographic projection

1749-1827 Pierre-Simon Laplace, French: Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism

1752-1833 Adrien-Marie Legendre, French: Abstract algebra, mathematical analysis, least squares method for curve-fitting and linear regression, quadratic reciprocity law, prime number theorem, elliptic functions

1768-1830 Joseph Fourier, French: Studied periodic functions and infinite sums in which the terms are trigonometric functions (Fourier series)

1777-1825 Carl Friedrich Gauss, German: Pattern in occurrence of prime numbers, construction of heptadecagon, Fundamental Theorem of Algebra, exposition of complex numbers, least squares approximation method, Gaussian distribution, Gaussian function, Gaussian error curve, non-Euclidean geometry, Gaussian curvature

1789-1857 Augustin-Louis Cauchy, French: Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory)

1790-1868 August Ferdinand Möbius, German: Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula

1791-1858 George Peacock, British: Inventor of symbolic algebra (early attempt to place algebra on a strictly logical basis)

1791-1871 Charles Babbage, British: Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer.

1792-1856 Nikolai Lobachevsky, Russian: Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai

1802-1829 Niels Henrik Abel, Norwegian: Proved impossibility of solving quintic equations, group theory, abelian groups, abelian categories, abelian variety

1802-1860 János Bolyai, Hungarian: Explored hyperbolic geometry and curved spaces independently of Lobachevsky

1804-1851 Carl Jacobi, German: Important contributions to analysis, theory of periodic and elliptic functions, determinants and matrices

1805-1865 William Hamilton, Irish: Theory of quaternions (first example of a non-commutative algebra)

1811-1832 Évariste Galois, French: Proved that there is no general algebraic method for solving polynomial equations of degree greater than four, laid groundwork for abstract algebra, Galois theory, group theory, ring theory, etc.

1815-1864 George Boole, British: Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science

1815-1897 Karl Weierstrass, German: Discovered a continuous function with no derivative, advancements in calculus of variations, reformulated calculus in a more rigorous fashion, pioneer in development of mathematical analysis

1821-1895 Arthur Cayley, British: Pioneer of modern group theory, matrix algebra, theory of higher singularities, theory of invariants, higher dimensional geometry, extended Hamilton's quaternions to create octonions

1826-1866 Bernhard Riemann, German: Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis

1831-1916 Richard Dedekind, German: Defined some important concepts of set theory such as similar sets and infinite sets, proposed Dedekind cut (now a standard definition of the real numbers)

1834-1923 John Venn, British: Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics)

1842-1899 Marius Sophus Lie, Norwegian: Applied algebra to geometric theory of differential equations, continuous symmetry, Lie groups of transformations

1845-1918 Georg Cantor, German: Creator of set theory, rigorous treatment of the notion of infinity and transfinite numbers, Cantor's theorem (which implies the existence of an "infinity of infinities”)

1848-1925 Gottlob Frege, German: One of the founders of modern logic, first rigorous treatment of the ideas of functions and variables in logic, major contributor to study of the foundations of mathematics

1849-1925 Felix Klein, German: Klein bottle (a one-sided closed surface in four-dimensional space), Erlangen Program to classify geometries by their underlying symmetry groups, work on group theory and function theory

1854-1912 Henri Poincaré, French: Partial solution to "three body problem," foundations of modern chaos theory, extended theory of mathematical topology, Poincaré conjecture

1858-1932 Giuseppe Peano, Italian: Peano axioms for natural numbers, developer of mathematical logic and set theory notation, contributed to modern method of mathematical induction

1861-1947 Alfred North Whitehead, British: Co-wrote "Principia Mathematica” (attempt to ground mathematics on logic)

1862-1943 David Hilbert, German: 23 "Hilbert problems," finiteness theorem, "Entscheidungsproblem" (decision problem), Hilbert space, developed modern axiomatic approach to mathematics, formalism

1864-1909 Hermann Minkowski, German: Geometry of numbers (geometrical method in multi-dimensional space for solving number theory problems), Minkowski space-time

1872-1970 Bertrand Russell, British: Russell's paradox, co-wrote "Principia Mathematica” (attempt to ground mathematics on logic), theory of types

1877-1947 G.H. Hardy, British: Progress toward solving Riemann hypothesis (proved infinitely many zeroes on the critical line), encouraged new tradition of pure mathematics in Britain, taxicab numbers

1878-1929 Pierre Fatou, French: Pioneer in field of complex analytic dynamics, investigated iterative and recursive processes

1881-1966 L.E.J. Brouwer, Dutch: Proved several theorems marking breakthroughs in topology (including fixed point theorem and topological invariance of dimension)

1887-1920 Srinivasa Ramanujan, Indian: Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions

1893-1978 Gaston Julia, French: Developed complex dynamics, Julia set formula

1903-1957 John von Neumann, Hungarian/American: Pioneer of game theory, design model for modern computer architecture, work in quantum and nuclear physics

1906-1978 Kurt Gödel, Austria: Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory

1906-1998 André Weil, French: Theorems allowed connections between algebraic geometry and number theory, Weil conjectures (partial proof of Riemann hypothesis for local zeta functions), founding member of influential Bourbaki group

1912-1954 Alan Turing, British: Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence

1913-1996 Paul Erdös, Hungarian: Set and solved many problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory

1917-2008 Edward Lorenz, American: Pioneer in modern chaos theory, Lorenz attractor, fractals, Lorenz oscillator, coined term "butterfly effect”

1919-1985 Julia Robinson, American: Work on decision problems and Hilbert's tenth problem, Robinson hypothesis

1924-2010 Benoît Mandelbrot, French: Mandelbrot set fractal, computer plottings of Mandelbrot and Julia sets

1928-2014 Alexander Grothendieck, French: Mathematical structuralist, revolutionary advances in algebraic geometry, theory of schemes, contributions to algebraic topology, number theory, category theory, etc.

1928-2015 John Nash, American: Work in game theory, differential geometry and partial differential equations, provided insight into complex systems in daily life such as economics, computing and military

1934-2007 Paul Cohen, American: Proved that continuum hypothesis could be both true and not true (i.e. independent from Zermelo-Fraenkel set theory)

1937- John Horton, Conway: British Important contributions to game theory, group theory, number theory, geometry and (especially) recreational mathematics, notably with the invention of the cellular automaton called the "Game of Life"

1947- Yuri Matiyasevich, Russian: Final proof that Hilbert's tenth problem is impossible (there is no general method for determining whether Diophantine equations have a solution)

1953- Andrew Wiles, British: Finally proved Fermat's Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves)

1966- Grigori Perelman, Russian: Finally proved Poincaré Conjecture (by proving Thurston's geometrization conjecture), contributions to Riemannian geometry and geometric topology

ne wouldthink that the multicultural agenda would penetrate and distort with such malignant deceptiveness the actual history of mathematics, a subject that bespeaks of exacting precision and truthfulness, but it has happened; academics have imposed upon all Western universities the requirement that this history must be responsive to "groups historically underrepresented in mathematics." This is not new. For some decades now multiculturalists have been rewriting European history in such a way that the "customs, heritage, history, and other aesthetic aspects" of non-European immigrants are incorporated as "essential components" of what they deceitfully call "an effective educational program." The just cited words are taken from, published back in 1991. Fast forward to 2017, and you will find that "multicultural mathematics" are now a major educational staple of the West. One of the authors of the books, George Gheverghese Joseph, professor at the University of Manchester, is also the author ofThis book has been very successful: first published by I.B. Tauris in 1991, by Penguin in 1992, then reprinted three times, and eventually released by Princeton University Press in 2000, with a third edition released in 2011. According to Google it has been cited over 800 times, with great reviews in such prestigious venues asand, and mostly five star reviews at Amazon. To top it all, this book has its own unique entry in Wikipedia.Yet,, in its effort to discredit the "Eurocentric" history of mathematics, is suffused with deceptive statements, misreadings of books, and wilful acts of historical fabrication. According to the author, the historical accounts of mathematics written by Europeans have exhibitedThis devaluation, the author says, was "part of the rationale for subjugation and dominance" of Third World peoples by Europeans (p. 4). Only two passages are offered by this Manchester academic as "a reasonable summary" of the bias and distortion of the history of mathematics by European historians.The first passage comes from a book published in 1908 by Rouse Ball:The second passage cited by Gheverghese comes from a book published in 1953 by Morris Kline:This statement by Kline, says Gheverghese, is wrong because itGheverghese's point is that, while the passage from 1908 may be excused, the view expressed in 1953 cannot be excused since much research was produced in the intervening years, making the Eurocentric model untenable. Gherverghese thinks it is time to propose a new model, and this is what he sets out to do, propose a model of the history of mathematics in which multiple cultures are shown to have played equally significant roles with "cross-fertilization between different mathematical traditions" happening at various points in time (pp. 5-9).We are supposed to believe that Gheverghese's model is complex, cosmopolitan, and nuanced, far superior to the simplistic, linear, one sided, parochial Eurocentric model. All multicultural revisionists make the same claims about their sophistication and their receptiveness to new research. Nonsense. The old research invalidates their claims, and the new research contradicts it., most of the books Gheverghese relies upon to construct his new model are authored by Europeans themselves, and the non-Europeans are either academics enjoying a great job inside a European university, or academics educated in European scholarly research., the book by Rouse Ball published in 1908 is actually very cognizant of the contribution of non-Europeans. The title of this book is PDF ).While the pages are few, the book begins with the following sections:Rouse Ball's point is not that the history of mathematics begin with the Greeks, but that it "cannot with certainty be traced back to any school or period before that of the Ionian Greeks." Why? Because only Europeans cared to write historical accounts of mathematics. To this day most of the histories of the achievements of non-Europeans have been written by Europeans., the very Gheverghese who complains about the "neglect of Arab contribution to... mathematics" (p. 344), did not tell his readers that Rouse Ball's book has two long chapters, IX and X, with the titles: "The Mathematics Of The Arabs" and "Introduction of Arabian Works into Europe, 1150-1450."Gherverghese, in his "final assessment" of the "Arab contribution," wants students to believe that he has proven wrong "the traditional view of the Arabs as mere custodians of Greek learning and transmitters of knowledge" by showing "how original were their contributions" to algebra and trigonometry, to the solution of cubic and of quadratic equations. Well, look at the table of contents of Rouse Ball's 400+ page book; it has sections on all these topics! He specifically affirms that Arabs were not mere transmitters of Greek knowledge but went beyond the Greeks. Here is a concluding passage:He writes about the limitations of their accomplishments only after carefully examining their contributions, which is normal, since the Arabs did not reach the level that modern Europeans would reach in subsequent centuries.Why would Gheverghese distort the scholarly contribution of Rouse Ball in this manner, assuming that no one would care to find out about its veracity? Because academics are committed to multiculturalism, and this ideology allows one to distort the truth, since it is based on false premises, and its aim is not scholarly but strictly ideological. The aim is to rewrite the history of Europeans in such a way that it is portrayed as a creation of multiple races so as to justify the arrival of masses of Africans, Asians, and Muslims., Gheverghese thinks he has a powerful argument against the Eurocentric model by drawing attention to the distinction between the Classical period of Greek science (600 to 300 BC) and the post-Alexandrian period (300 BC to 400 AD), and then arguing that the latter period was also Egyptian simply because the city of Alexandrian, which was the city where the science of this period was nurtured, was in Egypt. He writes that one of the "most striking features" of this period "was its cosmopolitanism — part Egyptian, part Greek, with a liberal sprinkling of Jews, Persians, Phoenicians and Babylonians" (p. 8).But there is an inconvenient fact about this claim: all the known scientists of Alexandria were Greek in ethnicity. The School of Alexandria is known as the greatest mathematical school of ancient times. The foremost names are: Euclid, Eratosthenes, Archimedes, Ptolemy, Apollonius, and many other less known names. Some few years ago, Lucio Russo made much of the contributions of the post-Alexandrian period, or the Hellenistic period, in his book(2004).The incredible contributions of the Hellenistic period have never been forgotten, in truth; every book I have read about the history of science and mathematics contains separate chapters (or sections) on this epoch, as an epoch. It has always been know that this was the period in which real science was born, rather than in the Classical Period, which was about natural philosophy.regarding Morris Kline, Gheverghese leaves out the fact that the title of Kline's book is, which is to say that this book is not intended as a history of mathematics; rather, as the opening sentence says:Since ancient Greek times, Europeans have been possessed by the idea that nature is characterized by regularity, design, lawful relationships and rational patterns, which the mind can know about, as a separate faculty of reasoning and objectivity, with a capacity for "pure thinking" in abstraction from bodily impulses and external contingencies, through the use of mathematics. This is why most of the history of mathematics is veritably a history of European accomplishments, which brings me to my last point of refutation., the notion that before the publication of Gheverghese's book the history of mathematics was under the tutelage of a Eurocentric model that ignored the contributions of other civilizations is preposterous. Why did he chose a book from 1908 and one from 1953 to make his case that non-Europeans mathematics has been neglected (not that the book from 1908, as I showed, neglected non-Europeans)? Why did he not cite passages from Carl Boyer's, first published in 1968? This book has chapters dedicated to "Egypt," and to "Mesopotamia," and to "China and India," and a chapter with the title "The Arabic Hegemony."Every book I have read and consulted on the history of mathematics has chapters on non-European contributions. In fact, we can go back to 1923 when D.E. Smith's book,, was published, to find two opening chapters on non-Europeans contributions, and also one chapter, plus half of another, on "Oriental" mathematics. There is also a separate section on the contributions of the Orient in all the chapters on Europeans.How can Gheverghese, and the many others promoting "multicultural mathematics," insist that Europeans neglected the contributions of others? Why do they keep repeating this when it is the other way around: Europeans were the only ones who wrote and acknowledged the contributions of others? The ideological reason is that multiculturalism is about enforcing cultural equality, about justifying the occupation of European lands by non-Europeans, and about giving the impression that non-Europeans are not only to be welcomed but have also been co-creators of the culture of Europeans.But if our concern is to write accurate histories of mathematics, we cannot but give far more priority to Europeans. Boyer, for example, has four chapters on non-Europeans, and 24 on Europeans, because Europeans did contribute a lot more to mathematics. That is just a fact, and if we are to write a proper history, Europeans must be paid more attention. But multiculturalism does not care about mathematics, about the truth; it cares about promoting other cultures inside the West and bringing about the destruction of Whites.D.E. Smith, I might add, co-authored a book on Japanese Mathematics , which goes to show that he cared about the contributions of non-Europeans, as has always been the case with Europeans, the authors of almost all the books on non-Europeans. The basic arguments that Smith presented in, in 1923, is still valid. He offered an opening chapter on Egypt, Mesopotamia, China, and India, as "pioneers in mathematical development before 1000 BC." Then he offered chapters on the contributions of the Classical and the Hellenistic Greeks, from 600 BC to 400 AD, roughly speaking. He did not have much to say about the Orient for this period simply because "so little was accomplished in the Orient from 1000 BC to 300 BC" (Dover ed., 1958, p. 95).He did recognize, however, that "the period from 300 BC to 500 AD was one of developmental activity in China" (p. 138); and that during the "five centuries from 500 to 1000 AD...Europe was intellectually dormant" (p. 148), and that during this same period "there were four or five mathematicians of prominence in India" (p. 152), as well as some prominent Chinese mathematicians. While it can be argued that he neglected the contributions of Muslims, and believed that the Arabs were "transmitters of learning rather than creators," he did offer sections on the "greatest mathematicians" during the Islamic ascendancy from the eight to the fourteenth century. He also said that, in China, the five centuries from 1000 to 1500 AD "were the most interesting" in the history of mathematics (p. 266), though he then qualified this overtly generous statement and restricted Chinese creativity to the thirteenth century only (p. 269).After 1200-1400, his focus shifts back to the Europeans and stays there because from this point on all the original ideas came from them alone. Smith takes a look at non-Europeans, and this is what he sees:Meanwhile, in the West we have one original mathematician after another from about the 1300s onward with only one guy from India. Here is a partial list up to the 1960s:The same pattern of immense European creativity can be observed in every other endeavour. Why this has been the case, why Europeans are responsible for almost everything great on the universe, requires an explanation. I have attempted some explanations inand, but more analysis and historical reflection is necessary, particularly in light of the fact that multicultural historians are trying to distort, hide, trash, and lie about these achievements.