Toán học Ấn Độ

Toán học Ấn Độ phát triển trên tiểu lục địa Ấn Độ^[1]^[2] từ 1200 TCN cho đến cuối thế kỷ 18. Trong thời kỳ cổ điển của toán học Ấn Độ (400 đến 1200), những cống hiến quan trọng được tạo ra bởi các học giả như là Aryabhata, Brahmagupta và Bhaskara II. hệ thống số dùng hệ thập phân^[3] được sử dụng ngày nay được ghi nhận đầu tiên trong toán học Ấn Độ.^[4] Toán học Ấn Độ có những cống hiến sớm trong nghiên cứu về số 0 như là một con số,^[5] số âm,^[6] số học và đại số.^[7] Thêm vào đó, lượng giác^[8] cũng đã phát triển tại Ấn Độ. Cụ thể, nền toán học này đã đưa ra những khái niệm hiện đại của sin và cosin.^[9] Những khái niệm toán học này đã được chuyển dịch đến Trung Đông, Trung Quốc và châu Âu^[7] và được phát triển xa đã định hình ra nhiều lĩnh vực của toán học ngày nay.

Những công trình toán học Ấn Độ thời cổ đại và trung cổ, đều được viết trong tiếng Phạn, thường bao gồm sutra trong một tập hợp của các quy tắc và vấn đề được xác định với một cơ cấu tốt trong thơ nhằm để hỗ trợ việc ghi nhớ bởi một học sinh. Điều đó được theo bởi một nhóm thứ hai bao gồm một bài bình luận bằng văn xuôi (thỉnh thoảng là những bình luận phức tạp được đưa ra bởi các học giả) giải thích vấn đề bằng nhiều chi tiết hơn và cung cấp sự biện hộ cho giải pháp của vấn đề đó. Trong phần văn xuôi này, cấu trúc (và vì thế sự ghi nhớ hóa của nó) không được xem xét là quá quan trọng như là các ý tưởng ở trong đó.^[1]^[10] Tất cả các công trình toán học đều được truyền miệng cho đến khoảng 500 TCN. Sau đó, chúng được truyền từ người này sang người khác bằng miệng và văn bản. Văn bản toán học mở rộng lâu đời nhất được sáng tác ở trên tiểu lục địa Ấn Độ là Bản Bakhshali viết trên vỏ cây cáng lò, được khám phá vào năm 1881 tại một ngôi làng tại Bakhshali, gần Peshawar (Pakistan hiện nay). Văn bản này có thể có niên đại vào thế kỷ 7.^[11]^[12]

Một bước ngoặt sau đó trong toán học Ấn Độ là sự phát triển của việc mở rộng theo chuỗi cho các công thức lượng giác (sin, cosin và arc tangent) bởi các nhà toán học của trường phái Kerala trong thế kỷ 15. Sự phát triển đáng chú ý này, được hoàn thành hai thế kỷ trước khi châu Âu phát minh ra calculus, cung cấp cái được xét như là ví dụ đầu tiên của một chuỗi năng lực (tách ra từ chuỗi hình học).^[13] Tuy nhiên, họ không công thức hóa một lý thuyết mang tính hệ thống của đạo hàm và tích phân, hoặc là không có băng chứng trực tiếp nào của kết quả của họ được truyền ra bên ngoài Kerala.^[14]^[15]^[16]^[17]

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Chú thích

^ ^a ^b Encyclopaedia Britannica (Kim Plofker) 2007, tr. 1
^ (Hayashi 2005, tr. 360–361)
^ Ifrah 2000, tr. 346: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own."
^ Plofker 2009, tr. 44–47
^ Bourbaki 1998, tr. 46: "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."
^ Bourbaki 1998, tr. 49: Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."
^ ^a ^b "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. ngày 16 tháng 5 năm 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."
^ (Pingree 2003, tr. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."
^ (Bourbaki 1998, tr. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."
^ Filliozat 2004, tr. 140–143
^ Hayashi 1995
^ Encyclopaedia Britannica (Kim Plofker) 2007, tr. 6
^ 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 generalised 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 Matthew 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. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented 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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Đọc thêm

Các cuốn sách viết bằng tiếng Phạn

Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages, ISBN 3-7643-7291-5.
Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages, ISBN 3-7643-7292-3.
Neugebauer, Otto; Pingree (eds.), David (1970), The Pañcasiddhāntikā of Varāhamihira, New edition with translation and commentary, (2 Vols.), CopenhagenQuản lý CS1: văn bản dư: danh sách tác giả (liên kết).
Pingree, David (ed) (1978), The Yavanajātaka of Sphujidhvaja, edited, translated and commented by D. Pingree, Cambridge, MA: Harvard Oriental Series 48 (2 vols.)Quản lý CS1: văn bản dư: danh sách tác giả (liên kết).
Sarma, K. V. (ed) (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, critically edited with Introduction and Appendices, New Delhi: Indian National Science AcademyQuản lý CS1: văn bản dư: danh sách tác giả (liên kết).
Sen, S. N.; Bag (eds.), A. K. (1983), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science AcademyQuản lý CS1: văn bản dư: danh sách tác giả (liên kết).
Shukla, K. S. (ed) (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara, critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science AcademyQuản lý CS1: văn bản dư: danh sách tác giả (liên kết).
Shukla, K. S. (ed) (1988), Āryabhaṭīya of Āryabhaṭa, critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with K.V. Sarma, New Delhi: Indian National Science AcademyQuản lý CS1: văn bản dư: danh sách tác giả (liên kết).

Liên kết ngoài

Science and Mathematics in India Lưu trữ 2010-07-28 tại Wayback Machine
An overview of Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2000.
Indian Mathematicians
Index of Ancient Indian mathematics Lưu trữ 2019-10-22 tại Wayback Machine, MacTutor History of Mathematics Archive, St Andrews University, 2004.
Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
Indian Mathematics trên chương trình In Our Time của BBC.
InSIGHT 2009 Lưu trữ 2006-02-25 tại Wayback Machine, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
Mathematics in ancient India by R. Sridharan
Combinatorial methods in ancient India
Mathematics before S. Ramanujan

Chủ đề Ấn Độ

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Nền tảng toán học \| Đại số \| Giải tích \| Hình học \| Lý thuyết số \| Toán học rời rạc \| Toán học ứng dụng \| Toán học giải trí \| Toán học tô pô \| Xác suất thống kê

[plofker-1] Encyclopaedia Britannica (Kim Plofker) 2007, tr. 1

[hayashi2005-p360-361-2] (Hayashi 2005, tr. 360–361)

[irfah346-3] Ifrah 2000, tr. 346: "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own."

[4] Plofker 2009, tr. 44–47

[bourbaki46-5] Bourbaki 1998, tr. 46: "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."

[bourbaki49-6] Bourbaki 1998, tr. 49: Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."

[concise-britannica-7] "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. ngày 16 tháng 5 năm 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."

[8] (Pingree 2003, tr. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."

[9] (Bourbaki 1998, tr. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."

[filliozat-p140to143-10] Filliozat 2004, tr. 140–143

[hayashi95-11] Hayashi 1995

[plofker-brit6-12] Encyclopaedia Britannica (Kim Plofker) 2007, tr. 6

[13] Stillwell 2004, tr. 173

[14] 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."

[15] 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 generalised 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"

[16] 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 Matthew 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."

[17] 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. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented 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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