Trường phái Kerala

Trường phái Kerala về Thiên văn học và Toán học
Trường phái Kerala về Thiên văn học và Toán học
Vị trí
	Kerala, Ấn Độ
Thông tin
Loại	Ấn Độ giáo, thiên văn học, toán học, khoa học
Sáng lập	Madhava của Sangamagrama

Trường phái Kerala, tên đầy đủ là Trường phái Kerala về Thiên văn học và Toán học là một trường phái nghiên cứu về thiên văn học và toán học được thành lập bởi Madhava của Sangamagrama, tại Kerala, Ấn Độ, với các thành viên: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri và Achyuta Panikkar. Trường phái này nở rộ từ thế kỷ 14 đến thế kỷ 16 và những khám phá nguyên bản của họ có vẻ kết thúc vào thời của Narayana Bhattathiri (1559-1632). Trong nỗ lực giải quyết các vấn đề thiên văn học, trường phái Kerala đã khám phá một cách độc lập một số các khái niệm toán học quan trọng. Những kết quả quan trọng nhất của họ - sự mở rộng hàng loạt cho các hàm số lượng giác - đã được miêu tả trong thơ bằng tiếng Phạn trong một cuốn sách được viết bởi Neelakanta có tên là Tantrasangraha, và được mô tả một lần nữa trong một bình luận về tác phẩm này, được gọi là Tantrasangraha-vakhya, của một tác giả vô danh. Các định lý được diễn tả mà không có chứng minh, các chứng minh hàng loạt cho sin, cosin và tang nghịch đảo được cung cấp vào một thế kỷ sau đó trong tác phẩm Yuktibhasa (khoảng năm 1500 - khoảng năm 1610), được viết bởi Jyesthadeva, bằng tiếng Malayalam, và trong một bình luận về Tantrasangraha.^[1]

Các tác phẩm của trường phái này, được hoàn thành trước hai thế kỷ trước sự ra đời của calculus, cung cấp cái mà bây giờ được xét đến như là ví dụ đầu tiên về chuỗi lũy thừa (tách biệt từ chuỗi hình học.^[2] Tuy nhiên, họ đã không công thức hóa một lý thuyết mang tính hệ thống về vi phân và tích phân, hoặc là không có bất kỳ bằng chứng trực tiếp nào về kết quả của học được chuyển ra bên ngoài Kerala.^[3]^[4]^[5]^[6]

Chú thích

^ 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, tr. 173)
^ (Bressoud 2002, tr. 12) Quote: "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 use."
^ Plofker 2001, tr. 293 Quote: "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"
^ Pingree 1992, tr. 562 Quote: "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, tr. 173–174 Quote: "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.
Bản mẫu:QuadThere 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."

Tham khảo

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.
Gupta, R. C. (1969) "Second Order of Interpolation of Indian Mathematics", Indian Journal of History of Science 4: 92-94
Hayashi, Takao (2003), “Indian Mathematics”, trong Grattan-Guinness, Ivor (biên tập), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 1, pp. 118-130, Baltimore, MD: The Johns Hopkins University Press, 976 pages, ISBN 0-8018-7396-7 Chú thích có tham số trống không rõ: |publication-year= (trợ giúp).
Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, ISBN 0-691-00659-8.
Katz, Victor J. (1995), “Ideas of Calculus in Islam and India”, Mathematics Magazine (Math. Assoc. Amer.), 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
Parameswaran, S. (1992) "Whish's showroom revisited", Mathematical Gazette 76, no. 475 pages 28–36
Pingree, David (1992), “Hellenophilia versus the History of Science”, Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257
Plofker, Kim (1996), “An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text”, Historia Mathematica, 23 (3): 246–256, doi:10.1006/hmat.1996.0026.
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.
Plofker, K. (ngày 20 tháng 7 năm 2007), “Mathematics of India”, trong Katz, Victor J. (biên tập), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (xuất bản 2007), tr. 385–514, ISBN 0-691-11485-4.
C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ', Philosophy East and West 51, University of Hawaii Press, 2001.
Roy, Ranjan (1990), “Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha”, Mathematics Magazine (Math. Assoc. Amer.), 63 (5): 291–306, doi:10.2307/2690896, JSTOR 2690896.
Sarma, K. V.; Hariharan, S. (1991). “Yuktibhasa of Jyesthadeva: a book of rationales in Indian mathematics and astronomy - an analytical appraisal”. Indian J. Hist. Sci. 26 (2): 185–207.
Singh, A. N. (1936), “On the Use of Series in Hindu Mathematics”, Osiris, 1: 606–628, doi:10.1086/368443, JSTOR 301627
Stillwell, John (2004), Mathematics and its History (ấn bản thứ 2), Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1.
Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610, vol. 2, Macerata, 1613.

Liên kết ngoài

The Kerala School, European Mathematics and Navigation, 2001.
An overview of Indian mathematics, MacTutor History of Mathematics archive, 2002.
Indian Mathematics: Redressing the balance, MacTutor History of Mathematics archive, 2002.
Keralese mathematics Lưu trữ 2018-05-29 tại Wayback Machine, MacTutor History of Mathematics archive, 2002.
Possible transmission of Keralese mathematics to Europe Lưu trữ 2018-01-03 tại Wayback Machine, MacTutor History of Mathematics archive, 2002.
"Indians predated Newton 'discovery' by 250 years" phys.org, 2007

[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.

[2] (Stillwell 2004, tr. 173)

[3] (Bressoud 2002, tr. 12) Quote: "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 use."

[4] Plofker 2001, tr. 293 Quote: "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"

[5] Pingree 1992, tr. 562 Quote: "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."

[6] Katz 1995, tr. 173–174 Quote: "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.
Bản mẫu:QuadThere 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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