케랄라 학파

케랄라 학파(Kerala school)는 중세 말기 인도의 타누르 왕국 케랄라 사람 이린나타필리 마드하반 남푸티리가 창시한 수학 및 천문학 학파다. 주요 소속 학자로 바타세리 파라메시바라 남부디리, 닐라칸타 소마야지, 제하데바, 아츄타 피샤라디, 멜파투르 나라야나 바타티리 등이 있다. 케랄라 학파는 14세기에서 16세기 사이에 번창하다가, 17세기의 나라야나 바타티리 이후로는 독창적 발견이 끊어졌다.

케랄라 학파는 천문학 문제들을 해결하기 위해 시도하는 과정에서 삼각함수의 급수전개를 비롯한 여러 가지 중요한 수학 개념들을 독립적으로 발견했다. 이런 내용들은 산스크리트어로 쓰인 『탄트라 상그라하』(1501년경)에 처음 언급되지만 이 책에는 증명이 없었다. 증명은 제하데바가 말레이어로 쓴 『육티바사』(1530년경)에서 제시되었다.^[1]

서양에서 미적분학이 발명되기 두 세기 전에 이루어진 케랄라 학파의 작업은 수학사상 최초로 멱급수를 사용한 사례로 인정되고 있다.^[2]^:173 그러나 케랄라 학파는 미분과 적분의 체계적 이론으로까지는 나아가지 못했고, 그들의 업적이 케랄라 바깥에 전파되어 유의미한 영향을 남겼다는 증거도 없다.^[3]^:12^[4]^:293^[5]^:562^[6]^:173–174

각주[편집]

↑ Roy, Ranjan. 1990. "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5): 291–306.
↑ Stillwell, John (2004), Mathematics and its History 2판, Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1 .
↑ Bressoud, David (2002), “Was Calculus Invented in India?”, The College Mathematics Journal (Math. Assoc. Amer.) 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972, There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much .
↑ Plofker, Kim (2001), “The "Error" in the Indian "Taylor Series Approximation" to the Sine”, Historia Mathematica 28 (4): 283–295, doi:10.1006/hmat.2001.2331, It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here |quote=에 라인 피드 문자가 있음(위치 1430) (도움말).
↑ Pingree, David (1992), “Hellenophilia versus the History of Science”, Isis 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164, One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.
↑ Plofker, K. (2007년 7월 20일), “Mathematics of India”, Katz, Victor J., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (2007에 출판됨), 385–514쪽, ISBN 978-0-691-11485-9, How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed.
　There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today. .

이 글은 수학에 관한 토막글입니다. 여러분의 지식으로 알차게 문서를 완성해 갑시다.

[roy-1] Roy, Ranjan. 1990. "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5): 291–306.

[Stillwell_2004-2] Stillwell, John (2004), Mathematics and its History 2판, Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1 .

[Bressoud_2002-3] Bressoud, David (2002), “Was Calculus Invented in India?”, The College Mathematics Journal (Math. Assoc. Amer.) 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972, There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much .

[Plofker_2001-4] Plofker, Kim (2001), “The "Error" in the Indian "Taylor Series Approximation" to the Sine”, Historia Mathematica 28 (4): 283–295, doi:10.1006/hmat.2001.2331, It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here |quote=에 라인 피드 문자가 있음(위치 1430) (도움말).

[Pingree_1992-5] Pingree, David (1992), “Hellenophilia versus the History of Science”, Isis 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164, One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.

[Katz_1995-6] Plofker, K. (2007년 7월 20일), “Mathematics of India”, Katz, Victor J., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (2007에 출판됨), 385–514쪽, ISBN 978-0-691-11485-9, How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed.
　There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today. .

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