???????

??????? ??????????????? ???????? ????????????? ??? ????? ???? ? ???????? ??????? ????????????? ?????????? ?????????? ＂???????????????????＂ ???????? ???????????????? Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[a]

百度这就需要欧盟及其成员国作出协调一致的坚定回应。

??????? (?????????: Mathematics) (?? ???? ????????????? ????????????) ??? ???????????????? ???????? ????? (quantity)^[?]? ????????? (structure)^[?]? ?????(space)^[?]? ????? ??????????? (change)^[?]???

???????????????? ??????? ????????????? ??????????????(conjectures) ?????? ???????? ????? ???????? ???????????????????? (patterns)??? ?????????????? ????? ????????????? ????? ??????????? ????? ???????????? ????? ???? ??????????? (mathematical proof)??? ??? ??????????????????? ?????????????????? ???????? ??????????? ??????????????????? ????????????? ???????? ??????????? ???????? ??? ??????? (abstraction) ????? ?????? (logic)? ?????????????? ????????? ??????????? ????? ?????? ??????????? ????? ????????? ????????? ????????????????? ????????????????? ??????????????????????? ?????????????????? ?????? ????????????? ????????????? ????????????????? ?????????? ????????????? ???????????????????? ????????????????? ????????????? ?????????? ?????? ????? ?????????????? ?????????? ????????? ??????????????

?????????? ??????????????? ??????? ???????? ??????????????? ??????????????????? ????????? ???????????????? ?????? ?????????????????????????? ??????????? ?????????????? ???????????? ????????????????? ?????? ??????? ????? ??????? ???? ?????????? ??? ??????? ????? ????????????? ?????????????????? ???????????????? ??????? ??????????? ???????????? ?????????????? ???????????? ????????????????????? ?????????? ?????????

????? ???? ??? ???????

??????? ???????? ?????????????? number (??????) ????? ????? ?????? ????? ?? ?? ?? ?? ??????????????????????? ??????? ??? ???????? ?????????????? mathematics ??????? (??) ??????? ????????????????? ???? ??? ???????? ???? ?????????? ????????????? ??? ????????????? ???????? ???????? ???? ??????????

???

????? ??????? ??????????????? ??????????? ??????????? ???? ????? ??????????? ? ??????????? ??? ? ??????????? The numerals used in the Bakhshali manuscript, dated between the 2nd century BCE and the 2nd century CE.

??? ???????? ????????????????? ???????????????????????????????????? ???? ????? ?????????? (abstractions) ??? ?????????? ???????????????????????? ?????? ???????????????????? ???????? ?????????????? ??????????????????? ????????????? ????????? ?????????????? ?????????????????? ????????????? ?????????? ??????????? ???????? ????????? ??????????? ????? ????????? ?????????????????? ????? ????????????? ???? ???????? ???????? ??? ???????? ?????????? ????? ?????????? ???????????? ????????????? ????? ??????????????????

????????? ?????????? ???????????????? ????? ??????? ??????????? ???????????????? ????? ????????????? ?????????????????? ????????? ?????????? ?????????? ????????? ????????? ^[?]

????????????? ????????????? ????????? ????????? ????? ??????? ????????? ???? ???? ????? ????????????????? ???????????????????????? ???????? ???????????????? ?????? ????????????????? ??????????????? ??????????????????? ???????????????? ????????? ????????? ??? ?????????? ????????????? ???????? ????????? (arithmetic) ??????????????? ??????? ????? ????? ???? ????? ??? ????????? (algebra) ???????????? ??????????????????? ??????????? (geometry) ????? ????? ????? ????????? ??????????? ???????? ???????????? ????????? ??????? ????? ????????????????^[?] ???????????????????????? ??????????? ???? ?????????????? ??? ?????????? ????????? ???? ????-???? ???? ??????? ??????????????????? ?????????? ??????????? (Pythagorean) ????? ????? ????? ???????????? ??????????????????? ????????? ???????????????? ?????????????? ??????????????? ????? ????????? ????? ?????????????? ?????? ?????????? ????????????? ?????????? ???????????????????? (????? ????? ????? ?)??? ????????????? ????? ????????? ????????? ????????? ?????????????? (archaeological record) ??? ???????????? ?????? ?????????????? (sexagesimal numeral system) ??????? ????????????? ????? ??????????????? ????????? ??????????? ????????????? ?????????????????? ^[?]

?????? ???? ? ?????????? ????????? ??????????? (Pythagorean) ????????????? ??????????? ??????????????????????? ??????????????????? ???????????????????^[?] ????? ??? ??????? ??????? ???????? (Euclid) ????????? ???? ?????????????? (axiomatic method) ???????? ????????? ?????????????????? ????? ??????????(axiom)? ?????????(theorem) ????? ????????????????????????? ?????????? Elements ??? ??? ??????????????????????????????? ???????? ????? ???????? ????????? ???????????????? ^[?] ??????????????? ?????????? ??????????????????? ???????????? ???????????? (c. 287 - 212 BC) ????? ??????? (Syracuse) ????????? ??? ^[??] ???????????? ????????????? ????????????? ??????????? ????????????????? ????? solids of revolution ????? method of exhaustion ??????????? ??????????????? ????? ???? ???? (arc of a parabola) ??????????????? ?????????? ???????????????? ?????????? (calculus)??? ^[??] ????????????? ???????????????????? ????????????????????? ??????????? ?????????????? ????????????? ???????? (Apollonius of Perga) ????? ???? ? ????????^[??] ?????????????? (Trigonometry) (?????????? Hipparchus of Nicaea ??????? ???????)? ?????????? (algebra) (?????????? Diophantus ?? ???? ? ???????)??^[??]

????????????????-?????? (Hindu-Arabic numeral system) ????? ?????????????????? ?????????? ????????? ????????? ????????????? ?????????? ?????????????????????? ?????????????????? ????? ?????????? ???????? ?????????? ??????????? ?????????????? ???????????????????? ?????????? ????????????????? ???? (sine) ??? ????? (cosine) ????? ??????? infinite series ?????????

??????????? ????????? ????????? ???? ? ????? ?? ?????????? ??????????????? ???????????????????????????? ???????????????? ??????????? ????????????????? ?????????????????? ???????????????? ????????? (algebra) ??? ???????????? ??????????????????????? ?????????? spherical trigonometry ??? ????? decimal point ??????????????????????? ?????????? ??????? ?????????????????? ??????? (??????) Al-Khwarismi, Omar Khayyam ????? Sharaf al-D?n al-Tūs? ??????

??????????? ????????????? ??????????????????????? ?????????????????????? ?????????? (calculus)??? Newton ??? Leibniz ??? ???????? ?????????????? ???? ?? ????????? Leonhard Euler ??? ???????? ?????????? ???????????? ???? ?? ???????? ?????????????????? ??????? ????? ???????????????? ???????? ????????????????????????? ?????????? ???????????? ???? ?? ?????????? ?????????????????????? Carl Friedrich Gauss ?????????? ????? ??????????????? ??????? ???????? ??????? ????????????????? ??????? ???????? ?????????? ??????????? (analysis)? ???????????????? ?????????? (differential geometry) ????????? ??????? (matrix theory)? ????????????? (number theory) ????? ??????? (statistics)?? ???? ?? ??????????? ??? Kurt G?del ????????? ?????????????? ????????????????????? incompleteness theorems ???????? ????????????? ?????? ??????????? ????????? ????? ??????? ???????? ??????? ???????????? ????????????????

?????????? ?????????????? ??????????????? ???????????????????? ??????? ??? ?????? ????? ??????????? ?????????? ?????????????? (????????????) ????????? ?????????????????? ?????????????? ?????????????????????? ?????? ????????????????????????????? ?????????? Mikhail B. Sevryuk ??????? ???????? ???????????????????????????? (Bulletin of the American Mathematical Society) ?????????? ????????????????? ???? ??????? ??????? ???? ?????????? ??????????? ???????? ?????????????? ????????? ??????????? ?????????? ????????? ????? ?.? ??????????????? ????????????? ??????????? ??????????????????? ??????????????? ??????????????????????????

????? ???????

Leonardo Fibonacci, ?????????? ???????? ?????????? ????????????????-?????? ??????? ???????? ? - ? ??????? ??????????????? the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World

?????????????? ??????? ???????????? ?????????????? ????????????? ??????????????? ^[??]^[??] ??????????(Aristotle) ???? ?????????????? ??????? ?????? ＂?????????? ???????? ?????????＂? ??????????????????? ???????? ??????????? ?? ?????????? ^[??] ???????? ?????? (Galileo Galilei) (????-????) ??? ＂??????????? ???????????????? ????????? ????? ?????? ???????????? ???????? ?????????????? ???????? ???????????? ????????? ?????? ???? ?????????????????? ???????? (?????)? ????? (triangles)? ??? (circles) ????? ????????????? (geometrical figures) ?????? ???????????? ?????? ????????????? ?????????????????? ?????? ??????????????? ???????????????? ???????? ????????? ?????? ??????????????????＂^[??] Carl Friedrich Gauss (???? - ????) ?????????????? ??????? ＂??????????? ???????????? ????????????????＂^[??] ????????? ??????????????? ?????????? ?????????????? ????????? ???????????????? ?????????? ?????????????????? ????????? ???? Benjamin Peirce (????- ????) ???? ??????? ＂?????? ?????? ????? ?????????????＂^[??] ??????? ?????????? (Albert Einstein)(????-????) ??? ＂???? ???????????????? ?????????????? ??????????? ?????????? ????????????????? ??????? ?????????? ?????????????????????? ?????????? ???????????? ??????????＂^[??]

???? ?? ??????? ???????? ????????????? ?????????????????????????? ??????????????? ??????????????? ?????????? ???????? ??????? ???? (group theory) ????? (projective geometry)??? ???????? ???? ????? (quantity) ??? ???? (measurement)??? ????????? ?????????? ??? ????????? ?????????? ?????????????????? ???????????????? ????????? ?????????? ?????????? ???????????????? ??????????? ????????? ????????????????????? ?????????????????? ?????????????? ?????????? ^[??] ??????????????? ?????? ?????? ?????????????????? ??????????? ????????????????? ??????????? ????????????? ??????? ????????? ?????????? ???????????????? ????????????????? ??????????????????????^[??]

??????? ????? ??????

?????????? ?????????? Carl Friedrich Gauss ??? ＂??????????? ???????????? ????????????????^[??] ???????????? ?????? (Marcus du Sautoy) ?????????? ＂?????????? ???????????? ?????? ... ??????????? ??????????????? ???????????＂??^[??]

?????? ??????? Fields of mathematics

?????????? ????????????? ???????? ???????????? ?????????? ???????? ?????? ?????????? ????? ????? ??????????? (???? ?????????? (arithmetic)? ????????(algebra)? ??????????(geometry)? ?????????? (analysis)?????? ??????????? ???????????????? ???????? ?????? ??????? (logic)? set theory (foundations)? ????????????????? ????????????????????? (applied mathematics) ????? ???????????????? (uncertainty) ???????? ????????????

????

↑ Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
↑ Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
↑ Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
↑ LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Calculus is the study of change—how things change, and how quickly they change.
↑ Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
↑ Kline 1990, Chapter 1.
↑ Boyer 1991, "Mesopotamia" p. 24–27.
↑ Heath, Thomas Little (1981) [originally published 1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. ISBN 978-0-486-24073-2.
↑ Boyer 1991, "Euclid of Alexandria" p. 119.
↑ Boyer 1991, "Archimedes of Syracuse" p. 120.
↑ Boyer 1991, "Archimedes of Syracuse" p. 130
↑ Boyer 1991, "Apollonius of Perga" p. 145.
↑ Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.
↑ Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.
↑ Tobies, Renate & Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN 978-3-0348-0229-1. [I]t is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
↑ Franklin, James (2025-08-04). Philosophy of Mathematics. p. 104. ISBN 9780080930589.
↑ Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and LeibnizArchived December 6, 2012, at the Wayback Machine., BBC Radio 4, September 27, 2010
↑ Waltershausen, p. 79
↑ Peirce, p. 97.
↑ Einstein, p. 28. The quote is Einstein's answer to the question: "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" This question was inspired by Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
↑ Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.
↑ Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.
↑ Waltershausen, p. 79
↑ du Sautoy, Marcus (June 25, 2010). "Nicolas Bourbaki". A Brief History of Mathematics. Event occurs at min. 12:50. BBC Radio 4. Archived from the original on December 16, 2016. Retrieved October 26, 2017.

[1] Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.

[2] Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. p. 4. ISBN 978-0-486-41712-7. Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.

[3] Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.

[4] LaTorre, Donald R.; Kenelly, John W.; Biggers, Sherry S.; Carpenter, Laurel R.; Reed, Iris B.; Harris, Cynthia R. (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. p. 2. ISBN 978-1-4390-4957-0. Calculus is the study of change—how things change, and how quickly they change.

[5] Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim

[6] Kline 1990, Chapter 1.

[7] Boyer 1991, "Mesopotamia" p. 24–27.

[8] Heath, Thomas Little (1981) [originally published 1921]. A History of Greek Mathematics: From Thales to Euclid. New York: Dover Publications. ISBN 978-0-486-24073-2.

[9] Boyer 1991, "Euclid of Alexandria" p. 119.

[10] Boyer 1991, "Archimedes of Syracuse" p. 120.

[11] Boyer 1991, "Archimedes of Syracuse" p. 130

[12] Boyer 1991, "Apollonius of Perga" p. 145.

[13] Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180.

[14] Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.

[15] Tobies, Renate & Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. p. 9. ISBN 978-3-0348-0229-1. [I]t is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.

[16] Franklin, James (2025-08-04). Philosophy of Mathematics. p. 104. ISBN 9780080930589.

[17] Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and LeibnizArchived December 6, 2012, at the Wayback Machine., BBC Radio 4, September 27, 2010

[18] Waltershausen, p. 79

[19] Peirce, p. 97.

[20] Einstein, p. 28. The quote is Einstein's answer to the question: "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" This question was inspired by Eugene Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

[21] Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.

[22] Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–385. doi:10.1007/BF01273907. JSTOR 3482762.

[23] Waltershausen, p. 79

[24] u Sautoy, Marcus (June 25, 2010). "Nicolas Bourbaki". A Brief History of Mathematics. Event occurs at min. 12:50. BBC Radio 4. Archived from the original on December 16, 2016. Retrieved October 26, 2017.

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