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↗ Philosophy & Its History ↗ World's Science & Technology History
📖《吴军数学通识讲义》吴军 吴文俊 主编 《中国数学史大系》
陶哲轩的《陶哲轩教你学数学》
Introduction to Mathematical Philosophy Bertrand Russell
📖 https://www.tedsundstrom.com/mathematical-reasoning-writing-and-proof Mathematical Reasoning，Writing and Proof Ted Sundstrom
- Note to Students
- Preface
- Table of Contents
- Writing Guidelines (Appendix A)
- Answers for the Progress Checks (Appendix B)
- Answers for Selected Exercises (Appendix C)
- Screencasts for Use with the Book
- Study Guides for Version 2.1
- Adoption List
- Resources for an Introduction to Proofs Course
- Reviews of the book on Amazon.com
- Reviews from the Open Textbook Library
wikipedia
https://en.wikipedia.org/wiki/Timeline_of_mathematics This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.
General
- Aaboe, Asger (1964). Episodes from the Early History of Mathematics. New York: Random House.
- Bell, E. T. (1937). Men of Mathematics. Simon and Schuster.
- Burton, David M. (1997). The History of Mathematics: An Introduction. McGraw Hill.
- Grattan-Guinness, Ivor (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press. ISBN 978-0-8018-7397-3.
- Kline, Morris. Mathematical Thought from Ancient to Modern Times.
- Struik, D. J. (1987). A Concise History of Mathematics, fourth revised edition. Dover Publications, New York.
Books on a specific period
- Gillings, Richard J. (1972). Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press.
- Heath, Thomas Little (1921). A History of Greek Mathematics. Oxford, Claredon Press.
- van der Waerden, B. L. (1983). Geometry and Algebra in Ancient Civilizations, Springer, ISBN 0-387-12159-5.
Books on a specific topic
- Corry, Leo (2015), A Brief History of Numbers, Oxford University Press, ISBN 978-0198702597
- Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion. ISBN 0-7868-6362-5.
- Menninger, Karl W. (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 978-0-262-13040-0.
- Stigler, Stephen M. (1990). The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press. ISBN 978-0-674-40341-3.
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for taxation, commerce, trade, and in astronomy, to record time and formulate calendars.
The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development, after basic arithmetic and geometry.
The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. The ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the 17th century and following discoveries of German mathematicians like Carl Friedrich Gauss and David Hilbert.
🔗 https://zh.wikipedia.org/zh-cn/%E6%95%B0%E5%AD%A6%E5%8F%B2#
数学史的主要研究对象是历史上的数学发现，调查它们的起源，或更广义地说，数学史就是对过去的数学方法与数学符号的探究。
数学起源于人类早期的生产活动，为古中国六艺之一，亦被古希腊学者视为哲学之起点。 数学最早用于人们计数、天文、度量甚至是贸易的需要。==这些需要可以简单地被概括为数学对结构、空间以及时间的研究；==对结构的研究是从数字开始的，首先是从我们称之为初等代数的——自然数和整数以及它们的算术关系式开始的。更深层次的研究是数论；对空间的研究则是从几何学开始的，首先是欧几里得几何和类似于三维空间[注 1]的三角学。后来产生了非欧几里得几何，在相对论中扮演着重要角色。
欧几里得所著《几何原本》中的一个证明 —— 被广泛认为是历史上最具影响力的教科书。[1] 在进入知识可以向全世界传播的现代社会以前，有记录的新数学发现仅仅在很少几个地区重见天日。目前最古老的数学文本是《普林顿 322》（古巴比伦，约公元前1900年[2]），《莱因德数学纸草书》（古埃及，约公元前2000年-1800年[3]），以及《莫斯科数学纸草书》（古埃及，约公元前1890年）。以上这些文本都涉及到了如今被称为毕达哥拉斯定理的概念，后者可能是继简单算术和几何后，最古老和最广泛传播的数学发现。
在公元前6世纪后，毕达哥拉斯将数学作为一门实证的学科进行研究，他创造了古希腊语单词μάθημα（mathema），意为“（被人们学习的）知识学问”[4]。希腊数学家在相当大的程度上改进了这些数学方法[注 2]，并扩大了数学的主题[5]。中国数学做了早期贡献，包括引入了位值制系统[6][7]。如今大行于世的印度-阿拉伯数字系统和运算方法，很可能是在公元后1000年的印度逐渐演化，并被伊斯兰数学家通过花拉子米的著作将其传到了西方[8][9]。伊斯兰数学则将以上这些文明的数学做了进一步的发展贡献。许多古希腊和伊斯兰数学著作随后被翻译成了拉丁文，引领了中世纪欧洲更深入的数学发展[10]。
从16世纪文艺复兴时期的意大利开始，算术、初等代数及三角学等初等数学已大体完备。17世纪变数概念的产生使人们开始研究变化中的量与量的互相关系和图形间的互相变换。随着自然科学和技术的进一步发展，为研究数学基础而产生的集合论和数理逻辑等也开始慢慢发展。
从古代到中世纪，数学发展的历史时期都伴随着数个世纪的停滞，但从16世纪以来，新的数学发展伴随新的科学发展，让数学不断加速大步前进，直至今日。
🔗 https://en.wikipedia.org/wiki/History_of_mathematics#Prehistoric
🔗 https://en.wikipedia.org/wiki/History_of_mathematics#Renaissance
During the Renaissance, the development of mathematics and of accounting were intertwined. While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.
Piero della Francesca (c. 1415–1492) wrote books on solid geometry and linear perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De quinque corporibus regularibus (On the Five Regular Solids).
Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494. It included a 27-page treatise on bookkeeping, "Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons. In Summa Arithmetica, Pacioli introduced symbols for plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.
In Italy, during the first half of the 16th century, Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.
Simon Stevin's De Thiende ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation in Europe, which influenced all later work on the real number system.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.
During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that were involved, were studied intensely
🔗 https://en.wikipedia.org/wiki/History_of_mathematics#19th_century
Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and quadratic reciprocity law.
This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces, and set the mathematical foundations for the theory of general relativity.
The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science. and Karl Weierstrass reformulated the calculus in a more rigorous fashion.
Also, for the first time, the limits of mathematics were explored. Paolo Ruffini, Niels Henrik Abel, and Évariste Galois proved there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve these problems since the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.
In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L.E.J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.
The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865,) the Société mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883,) and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Association, was formed in 1899, in the context of a vector controversy. In 1897, Kurt Hensel introduced p-adic numbers.
[!links] ↗ Mathematical Logic (Foundations of Mathematics) ↗ Formal System, Formal Logics, and Its Semantics
🔗 https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis
The foundational crisis of mathematics arose at the end of the 19th century and the beginning of the 20th century with the discovery of several paradoxes or counter-intuitive results.
The first one was the proof that the parallel postulate cannot be proved. This results from a construction of a non-Euclidean geometry inside Euclidean geometry, whose inconsistency would imply the inconsistency of Euclidean geometry. A well known paradox is Russell's paradox, which shows that the phrase "the set of all sets that do not contain themselves" is self-contradictory. Other philosophical problems were the proof of the existence of mathematical objects that cannot be computed or explicitly described, and the proof of the existence of theorems on natural numbers that cannot be proved with Peano arithmetic (see Gödel's incompleteness theorems).
Several schools of philosophy of mathematics were challenged with these problems in the 20th century, and are described below.
==These problems were also studied by mathematicians, and this led to establish mathematical logic as a new area of mathematics, consisting of providing mathematical definitions to logics (sets of inference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts.== The Principia Mathematica is a landmark result in mathematical logic and foundations published by Russell and Alfred North Whitehead in 1913.
Mathematical logic led to unexpected results, such as Gödel's incompleteness theorems, which, roughly speaking, assert that, if a theory contains the standard arithmetic, it cannot be used to prove that it itself is not self-contradictory; and, if it is not self-contradictory, there are theorems that cannot be proved inside the theory, but are nevertheless true in some technical sense.
Zermelo–Fraenkel set theory with the axiom of choice (ZFC) is a logical theory established by Ernst Zermelo and Abraham Fraenkel. It became the standard foundation of modern mathematics, and, unless the contrary is explicitly specified, it is used in all modern mathematical texts, generally implicitly.
Simultaneously, the axiomatic method became a de facto standard: the proof of a theorem must result from explicit axioms and previously proved theorems by the application of clearly defined inference rules. The axioms need not correspond to some reality. Nevertheless, it is an open philosophical problem to explain why the axiom systems that lead to rich and useful theories are those resulting from abstraction from the physical reality or other mathematical theory.
In summary, the foundational crisis is essentially resolved, and this opens new philosophical problems. In particular, it cannot be proved that the new foundation (ZFC) is not self-contradictory. It is a general consensus that, if this would happen, the problem could be solved by a mild modification of ZFC.
🔗 https://en.wikipedia.org/wiki/History_of_mathematics#21st_century
In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems. In 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award, as he was critical of the mathematics establishment).
Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive toward open access publishing, first made popular by arXiv.
Many other important problems have been solved in this century. Examples include the Green–Tao theorem (2004), existence of bounded gaps between arbitrarily large primes (2013), and the modularity theorem (2001). The AKS primality test was published in 2002, which is the first algorithm that can determine whether a number is prime or composite in polynomial time. A proof of Goldbach's weak conjecture was published by Harald Helfgott in 2013; as of 2025, the proof has not yet been fully reviewed. The first einstein was discovered in 2023.
In addition, a lot of work has been done toward long-lasting projects which began in the twentieth century. For example, the classification of finite simple groups was completed in 2008. Similarly, work on the Langlands program has progressed significantly, and there have been proofs of the fundamental lemma (2008), as well as a proposed proof of the geometric Langlands correspondence in 2024.
🔗 https://zh.wikipedia.org/wiki/%E4%B8%AD%E5%9C%8B%E6%95%B8%E5%AD%B8%E5%8F%B2
中國數學史是指中國的數學發展史。中國傳統數學稱為算學，起源于仰韶文化，距今有五千余年历史，在周公时代，数乃是六艺之一。在春秋时代十进位制的筹算已经普及。著名日本数学史家三上义夫指出，中国算学的发展有二三千年之久，如此长久的发展历史，世界各国未曾有过，希腊自公元前6世纪到公元4世纪，仅一千年历史；阿拉伯数学限于公元8世纪到13世纪。“中国之算学史，其有长期之发展，不能不谓之为世界中稀有之例也
