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Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics).
数学是一个发现和组织为经验科学和数学本身的需要而发展和证明的方法、理论和定理的研究领域。数学有很多领域，包括数论（对数字的研究）、代数（对公式和相关结构的研究）、几何（对形状和包含它们的空间的研究）、分析（对连续变化的研究） ，和集合论（目前用作所有数学的基础）。

Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.^[1]
数学涉及对抽象对象的描述和操作，这些抽象对象由自然的抽象组成，或者在现代数学中由规定具有某些属性（称为公理）的纯粹抽象实体组成。数学使用纯粹的理性来证明对象的属性，这种证明包括对已经确定的结果一系列演绎规则的应用。这些结果包括先前证明的定理、公理，以及（在从自然中抽象的情况下）一些基本属性，这些属性被认为是所考虑的理论的真正起点。1

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics) but often later find practical applications.^[2][3]
数学在自然科学、工程、医学、金融、计算机科学和社会科学中都是必不可少的。尽管数学广泛用于对现象进行建模，但数学的基本真理独立于任何科学实验。数学的某些领域，例如统计学和博弈论，其发展与其应用密切相关，并且通常被归为应用数学。其他领域的发展独立于任何应用（因此被称为纯数学），但通常后来才找到实际应用。 23

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.^[4] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra^[a] and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.^[5] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,^[6] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.
从历史上看，证明的概念及其相关的数学严谨性首先出现在希腊数学中，尤其是在欧几里得的《几何原本》中。4从一开始，数学就主要分为几何和算术（自然数和分数的运算），直到16 世纪和 17 世纪，代数和微积分作为新领域被引入。从那时起，数学创新和科学发现之间的相互作用导致两者的发展同步增长。5 19 世纪末，数学的基础危机导致了公理化方法的系统化6，这预示着数学领域及其应用领域的数量急剧增加。当代数学学科分类列出了六十多个一级数学领域。

The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt",^[7] "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times.^[b] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".^[11] In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art".^[7]
数学一词源自古希腊语máthēma ( μάθημα )，意思是“所学的东西”，7“人们所了解的东西”，因此也有“研究”和“科学”的意思。即使在古典时代，这个词也具有“数学研究”的更狭义和更专业的含义。b 它的形容词是mathēmatikós ( μαθηματικός )，意思是“与学习有关”或“勤奋的”，同样进一步意味着“数学的” .11 特别是， mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ；拉丁语： ars mathematica ) 意思是“数学艺术”。7

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.^[12]
同样，毕达哥拉斯主义的两个主要思想流派之一被称为mathēmatikoi (μαθηματικοί)——当时的意思是“学习者”，而不是现代意义上的“数学家”。毕达哥拉斯学派可能是第一个将这个词的使用限制为仅用于算术和几何研究的人。到亚里士多德时代（公元前 384-322 年），这一含义已完全确立。 12

In Latin and English, until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.^[13]
在拉丁语和英语中，直到 1700 年左右，数学一词更常指“占星学”（有时也指“天文学”），而不是“数学”；大约从1500年到1800年，这个含义逐渐变为现在的含义。这种变化导致了一些误译：例如，圣奥古斯丁警告基督徒应该提防mathematici ，意思是“占星家”，有时被误译为对数学家的谴责.13

The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek.^[14] In English, the noun mathematics takes a singular verb. It is often shortened to maths^[15] or, in North America, math.^[16]
英语中明显的复数形式可以追溯到拉丁语中性复数mathematica （西塞罗），它基于希腊语复数ta mathēmatiká （ τὰμαθηματικά ），大致意思是“所有数学的东西”，尽管英语似乎只借用了形容词mathematic（ al)并按照继承自希腊语的物理学和形而上学的模式重新形成了名词math。14在英语中，名词math采用单数动词。它通常缩写为maths 15，或者在北美缩写为math 0.16

Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes.^[17] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.^[18]
在文艺复兴之前，数学分为两个主要领域：算术（关于数字的处理）和几何（关于形状的研究）。 17 某些类型的伪科学，例如命理学和占星学，当时并没有与数学明确区分开来。 18

During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas—arithmetic, geometry, algebra, calculus^[19]—endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics.^[20] The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.^[21]
文艺复兴时期，又出现了两个领域。数学符号导致了代数，粗略地说，代数包括公式的研究和操作。微积分由微分微积分和积分两个子领域组成，是对连续函数的研究，它对由变量表示的变化量之间的典型非线性关系进行建模。这种划分为四个主要领域——算术、几何、代数、微积分19——一直持续到 19 世纪末。天体力学和固体力学等领域当时由数学家研究，但现在被认为属于物理学。20组合数学学科在历史上的大部分时间里都得到了研究，但直到 17 世纪才成为数学的一个独立分支。 .21

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.^[22][6] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.^[23] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.^[24]
19世纪末，数学的基础危机和由此产生的公理化方法的系统化导致了数学新领域的爆发。226 2020年数学学科分类包含不少于63个一级领域。23其中一些领域对应于旧的部门，就像数论（高等算术的现代名称）和几何一样。其他几个第一级区域的名称中包含“几何”，或者通常被认为是几何的一部分。代数和微积分不作为第一级区域出现，而是分别分为多个第一级区域。其他第一级领域出现在 20 世纪或以前不被视为数学，例如数理逻辑和基础.24

Number theory began with the manipulation of numbers, that is, natural numbers ${\displaystyle (\mathbb {N} ),}$ and later expanded to integers ${\displaystyle (\mathbb {Z} )}$ and rational numbers ${\displaystyle (\mathbb {Q} ).}$ Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.^[25] Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.^[26] The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.^[27]
数论始于对数字（即自然数）的操纵 ${\displaystyle (\mathbb {N} ),}$ 后来扩展到整数 ${\displaystyle (\mathbb {Z} )}$ 和有理数 ${\displaystyle (\mathbb {Q} ).}$ 数论曾经被称为算术，但现在这个术语主要用于数值计算。25 数论的历史可以追溯到古巴比伦，可能还有中国。两位著名的早期数论学家是古希腊的欧几里得和亚历山大的丢番图。26 现代数论研究的抽象形式主要归功于皮埃尔·德·费马和莱昂哈德·欧拉。在Adrien-Marie Legendre和Carl Friedrich Gauss的贡献下，该领域取得了圆满成果 .27

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra.^[28] Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.^[29]
许多容易表述的数字问题的解决方案需要复杂的方法，通常来自数学领域。一个突出的例子是费马大定理。这个猜想由 Pierre de Fermat 于 1637 年提出，但直到 1994 年才被Andrew Wiles证明，他使用了代数几何的图式论、范畴论和同调代数等工具。28 另一个例子是哥德巴赫猜想，它断言每个大于2的偶数是两个素数之和。克里斯蒂安·哥德巴赫 (Christian Goldbach)于 1742 年提出，尽管付出了巨大努力，但仍未得到证实。29

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).^[24]
数论包括几个子领域，包括解析数论、代数数论、数几何（面向方法）、丢番图方程和超越论（面向问题）24

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.^[30]
几何是数学最古老的分支之一。它始于有关形状（例如直线、角度和圆形）的经验配方，这些形状主要是为了测量和建筑的需要而开发的，但后来已经扩展到许多其他子领域。 30

A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.^[31][32]
一个根本性的创新是古希腊人引入了证明的概念，它要求每个断言都必须得到证明。例如，仅通过测量来验证两个长度是否相等是不够的；它们的相等性必须通过先前接受的结果（定理）和一些基本陈述的推理来证明。基本陈述不需要证明，因为它们是不言而喻的（假设），或者是研究主题定义的一部分（公理）。这一原理是所有数学的基础，首先针对几何学进行了阐述，并由欧几里得于公元前 300 年左右在他的《几何原本》一书中系统化。3132

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.^[c][30]
由此产生的欧几里德几何是对由欧几里德平面（平面几何）和三维欧几里德空间中的线、平面和圆构成的形状及其排列的研究。c30

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.^[33]
欧几里得几何的发展没有改变方法或范围，直到 17 世纪，笛卡尔引入了现在所谓的笛卡尔坐标。这构成了范式的重大变化：它不再将实数定义为线段的长度（参见数轴），而是允许使用点的坐标（即数字）来表示点。因此，代数（以及后来的微积分）可以用来解决几何问题。几何学被分为两个新的子领域：合成几何学（使用纯几何方法）和解析几何学（系统地使用坐标）33。

Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.^[30]
解析几何允许研究与圆和直线无关的曲线。这种曲线可以定义为函数图，对函数图的研究导致了微分几何。它们也可以被定义为隐式方程，通常是多项式方程（它催生了代数几何）。解析几何还使得考虑高于三个维度的欧几里得空间成为可能。 30

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.^[34][6] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space.^[35]
十九世纪，数学家发现了不遵循平行公设的非欧几里得几何。通过质疑该假设的真实性，这一发现被视为与罗素悖论一起揭示了数学的基本危机。这方面的危机是通过公理化方法的系统化解决的，并承认所选公理的真实性不是数学问题。346反过来，公理化方法允许研究通过改变公理或通过改变公理获得的各种几何形状。考虑在空间的特定变换下不会改变的属性 .35

Today's subareas of geometry include:^[24]
今天的几何子领域包括：24

• Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
  射影几何由吉拉德·笛沙格 (Girard Desargues)在 16 世纪提出，通过在平行线相交的无穷远处添加点来扩展欧几里得几何。通过统一相交线和平行线的处理，简化了经典几何的许多方面。
• Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
  仿射几何，研究与平行性相关且独立于长度概念的性质。
• Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions.
  微分几何，对曲线、曲面及其推广的研究，使用可微函数定义。
• Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
  流形理论，研究不一定嵌入更大空间的形状。
• Riemannian geometry, the study of distance properties in curved spaces.
  黎曼几何，弯曲空间中距离特性的研究。
• Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.
  代数几何，研究曲线、曲面及其推广，使用多项式定义。
• Topology, the study of properties that are kept under continuous deformations.
  拓扑学，研究连续变形下的特性。
  • Algebraic topology, the use in topology of algebraic methods, mainly homological algebra.
    代数拓扑，拓扑中使用的代数方法，主要是同调代数。
• Discrete geometry, the study of finite configurations in geometry.
  离散几何，几何中有限构型的研究。
• Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.
  凸几何，凸集的研究，其重要性源于其在优化中的应用。
• Complex geometry, the geometry obtained by replacing real numbers with complex numbers.
  复几何，用复数代替实数得到的几何。

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.^[37][38] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.^[39] Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side.^[40] The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise.^[41][42]
代数是操纵方程和公式的艺术。丢番图（公元 3 世纪）和花剌子米（公元 9 世纪）是代数的两个主要先驱。3738 丢番图通过推导新的关系来解决一些涉及未知自然数的方程，直到获得解。39 花剌子米引入了变换方程的系统方法，例如将一项从方程的一侧移到另一侧。40 代数一词源自阿拉伯语单词al-jabr，意思是“破碎部分的重聚”，他在标题中用它来命名这些方法之一他的主要论文.4142

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.^[43] Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.^[44]
只有弗朗索瓦·维埃特（François Viète，1540-1603）才使代数成为一个独立的领域，他引入了使用变量来表示未知或未指定的数字。 43变量允许数学家描述必须对使用数学表示的数字进行的运算。公式.44

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.^[45] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.^[46]
直到19世纪，代数主要包括线性方程（现在的线性代数）和单个未知数的多项式方程的研究，这些方程被称为代数方程（这个术语仍在使用，尽管可能含糊不清）。在 19 世纪，数学家开始使用变量来表示数字以外的事物（例如矩阵、模整数和几何变换），算术运算的推广通常是有效的。45代数结构的概念解决了这个问题，包括元素未指定的集合、作用于集合元素的操作以及这些操作必须遵循的规则。因此，代数的范围扩大到包括代数结构的研究。这个代数对象被称为现代代数或抽象代数，这是由艾美·诺特的影响和著作建立的。46

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:^[24]
在数学的许多领域中，某些类型的代数结构具有有用且通常是基本的属性。他们的研究成为代数的独立部分，包括：24

• group theory;
  群论；
• field theory;
  场论；
• vector spaces, whose study is essentially the same as linear algebra;
  向量空间，其研究本质上与线性代数相同；
• ring theory; 环理论；
• commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
  交换代数，即交换环的研究，包括多项式的研究，是代数几何的基础部分；
• homological algebra;
  同调代数；
• Lie algebra and Lie group theory;
  李代数和李群理论；
• Boolean algebra, which is widely used for the study of the logical structure of computers.
  布尔代数，广泛用于研究计算机的逻辑结构。

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.^[47] The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.^[48]
将代数结构的类型作为数学对象进行研究是普适代数和范畴论的目的。47后者适用于所有数学结构（不仅是代数结构）。它最初与同调代数一起被引入，以允许对非代数对象（例如拓扑空间）进行代数研究；这个特定的应用领域称为代数拓扑.48

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.^[49] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.^[50] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.^[51]
微积分，以前称为无穷小微积分，由 17 世纪数学家牛顿和莱布尼茨独立且同时引入。49 从根本上讲，它是对相互依赖的变量关系的研究。欧拉在 18 世纪扩展了微积分，引入了函数的概念和许多其他成果。50 目前，“微积分”主要指该理论的基本部分，“分析”通常用于高级部分。 51

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:^[24]
分析进一步分为实数分析（其中变量代表实数）和复数分析（其中变量代表复数） 。分析包括许多其他数学领域共享的子领域，其中包括：24

• Multivariable calculus 多变量微积分
• Functional analysis, where variables represent varying functions;
  功能分析，其中变量代表不同的功能；
• Integration, measure theory and potential theory, all strongly related with probability theory on a continuum;
  积分、测度论和势论，都与连续统上的概率论密切相关；
• Ordinary differential equations;
  常微分方程；
• Partial differential equations;
  偏微分方程；
• Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.
  数值分析，主要致力于在计算机上计算许多应用中出现的常微分方程和偏微分方程的解。

Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.^[52] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.^[d] Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.^[53]
从广义上讲，离散数学是对单个、可数数学对象的研究。一个例子是所有整数的集合。52因为这里的研究对象是离散的，所以微积分和数学分析的方法并不直接适用。d算法——尤其是它们的实现和计算复杂性——在离散数学中发挥着重要作用。53

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.^[54] The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.^[55]
四色定理和最优球体堆积是 20 世纪下半叶解决的离散数学的两个主要问题。 54 至今仍然开放的P 与 NP 问题对于离散数学也很重要，因为它的解决方案将可能会影响大量计算困难的问题。55

Discrete mathematics includes:^[24]
离散数学包括：24

• Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
  组合学，枚举满足某些给定约束的数学对象的艺术。最初，这些对象是给定集合的元素或子集；这已扩展到各种对象，从而在组合数学和离散数学的其他部分之间建立了牢固的联系。例如，离散几何包括计算几何形状的配置
• Graph theory and hypergraphs
  图论和超图
• Coding theory, including error correcting codes and a part of cryptography
  编码理论，包括纠错码和密码学的一部分
• Matroid theory
  拟阵理论
• Discrete geometry 离散几何
• Discrete probability distributions
  离散概率分布
• Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
  博弈论（虽然也研究了连续博弈，但大多数常见游戏，例如国际象棋和扑克都是离散博弈）
• Discrete optimization, including combinatorial optimization, integer programming, constraint programming
  离散优化，包括组合优化、整数规划、约束规划

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.^[56][57] Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.^[58]
数理逻辑和集合论这两门学科自19世纪末以来就属于数学。5657在此时期之前，集合不被认为是数学对象，而逻辑虽然用于数学证明，但属于哲学而不是哲学。由数学家专门研究58

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets^[59] but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.^[60] In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour.^[61]
在康托研究无限集合之前，数学家们不愿意考虑实际上的无限集合，并认为无穷是无休止枚举的结果。康托尔的工作冒犯了许多数学家，不仅因为考虑了实际上无限的集合59，而且根据康托尔的对角线论证，这表明这意味着无穷大的不同大小。这引发了对康托集合论的争议.60 在同一时期，数学的各个领域都得出结论，以前对基本数学对象的直观定义不足以确保数学的严谨性.61

This became the foundational crisis of mathematics.^[62] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.^[22] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.^[63] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.^[64]
这成为数学的根本危机。62最终通过在形式化集合论中系统化公理化方法，在主流数学中解决了这个问题。粗略地说，每个数学对象都是由所有相似对象的集合以及这些对象必须具有的属性来定义的。22例如，在皮亚诺算术中，自然数是通过“零是一个数字”、“每个数字都有一个唯一的后继”，“除了零之外的每个数字都有一个唯一的前任”，以及一些推理规则。63这种对现实的数学抽象体现在现代形式主义哲学中，由大卫·希尔伯特在1910年左右创立。64

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.^[65] This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.^[66][67]
以这种方式定义的对象的“本质”是数学家留给哲学家的一个哲学问题，即使许多数学家对此本质有看法，并使用他们的观点（有时称为“直觉”）来指导他们的研究和证明。该方法允许将“逻辑”（即允许的演绎规则集）、定理、证明等视为数学对象，并证明关于它们的定理。例如，哥德尔的不完备性定理断言，粗略地说，在每个包含自然数的一致形式系统中，都有一些定理是正确的（在更强的系统中是可证明的），但在系统内部是不可证明的。 65 这种方法数学基础在20世纪上半叶受到以布劳威尔为首的数学家的挑战，他们提倡直觉逻辑，而直觉逻辑明显缺乏排中律.6667

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory.^[24] Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, formal verification, program analysis, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.^[68]
这些问题和争论导致了数理逻辑的广泛扩展，包括模型论（在其他理论中对某些逻辑理论进行建模）、证明论、类型论、可计算性理论和计算复杂性理论等子领域。24尽管数理逻辑的这些方面在计算机兴起之前就被引入，它们在编译器设计、形式验证、程序分析、证明助手和计算机科学的其他方面的使用反过来又促进了这些逻辑理论的扩展。 68

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.^[70]
统计学领域是一种数学应用，用于使用基于数学方法（尤其是概率论）的程序来收集和处理数据样本。统计学家通过随机抽样或随机实验生成数据 0.70

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.^[71] Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.^[72]
统计理论研究决策问题，例如最小化统计操作的风险（预期损失），例如使用参数估计、假设检验和选择最佳程序等过程。在数理统计的这些传统领域中，统计决策问题是通过在特定约束下最小化目标函数（例如预期损失或成本）来制定的。例如，设计一项调查通常涉及在给定的置信水平下最小化估计总体平均值的成本。71由于使用了优化，统计数学理论与其他决策科学重叠，例如运筹学、控制论、和数理经济学.72

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.^[73][74] Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors.^[75] Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.
计算数学是对对于人类数值能力来说通常太大的数学问题的研究。7374数值分析研究使用泛函分析和近似理论分析问题的方法；数值分析广泛地包括近似和离散化的研究，特别关注舍入误差。75 数值分析，更广泛地说，科学计算还研究数学科学的非分析主题，特别是算法矩阵和图论。计算数学的其他领域包括计算机代数和符号计算。

In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.^[76][77] Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[78] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC.^[79] Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.^[80]
除了认识如何计算物理对象之外，史前人民可能还知道如何计算抽象数量，例如时间（日、季节或年）。7677 直到公元前3000 年左右，更复杂的数学的证据才出现，当时巴比伦人和埃及人开始使用算术、代数和几何进行税收和其他金融计算、建筑和建造以及天文学。 78美索不达米亚和埃及最古老的数学文本是公元前 2000 年至 1800 年。 79 许多早期文本提到了毕达哥拉斯三元组，因此由此推论，毕达哥拉斯定理似乎是继基本算术和几何之后最古老、最广泛的数学概念。正是在巴比伦数学中，基本算术（加法、减法、乘法和除法）首次出现在考古记录中。巴比伦人还拥有位值系统并使用六十进制数字系统，该系统至今仍在用于测量角度和时间。 80

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.^[81] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.^[82] His book, Elements, is widely considered the most successful and influential textbook of all time.^[83] The greatest mathematician of antiquity is often held to be Archimedes (c. 287 – c. 212 BC) of Syracuse.^[84] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.^[85] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),^[86] trigonometry (Hipparchus of Nicaea, 2nd century BC),^[87] and the beginnings of algebra (Diophantus, 3rd century AD).^[88]
在公元前 6 世纪，希腊数学开始作为一门独特的学科出现，一些古希腊人，如毕达哥拉斯学派，似乎认为它本身就是一门学科。81 公元前 300 年左右，欧几里得通过公设和第一章的方式组织了数学知识。原理，演变成当今数学中使用的公理方法，包括定义、公理、定理和证明。82 他的书《几何原本》被广泛认为是有史以来最成功和最有影响力的教科书。83古代通常被认为是锡拉丘兹的阿基米德（约公元前287 – 约公元前 212 年）。84 他开发了计算旋转固体的表面积和体积的公式，并使用穷举法计算圆弧下的面积抛物线与无限级数的求和，其方式与现代微积分没有太大不同。 85 希腊数学的其他显着成就是圆锥曲线（佩尔加的阿波罗尼乌斯，公元前 3 世纪），86三角学（尼西亚的喜帕恰斯，公元前 2 世纪） ,87 和代数的开端（丢番图，公元 3 世纪）.88

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.^[89] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.^[90][91]
当今世界各地使用的印度-阿拉伯数字系统及其运算使用规则，在印度的公元第一个千年中不断演变，并通过伊斯兰数学传播到西方世界。89 其他值得注意的发展印度数学包括正弦和余弦的现代定义和近似，以及无穷级数的早期形式.9091

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.^[92] Many notable mathematicians from this period were Persian, such as Al-Khwarizmi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.^[93] The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.^[94]
在伊斯兰教的黄金时代，特别是在 9 世纪和 10 世纪，数学在希腊数学的基础上出现了许多重要的创新。伊斯兰数学最显着的成就是代数的发展。伊斯兰时期的其他成就包括球面三角学的进步以及在阿拉伯数字系统中添加小数点。 92 这一时期的许多著名数学家都是波斯人，例如花剌子米、奥马尔·海亚姆和沙拉夫·阿尔丁·塔西.93 希腊语和阿拉伯语数学文本在中世纪又被翻译成拉丁语并在欧洲提供。 94

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.^[95]
在近代早期，数学在西欧开始加速发展，其创新彻底改变了数学，例如弗朗索瓦·维特（François Viète，1540-1603）引入变量和符号表示法，约翰·纳皮尔（John Napier）在 1603 年引入对数。 1614 年，它大大简化了数值计算，特别是天文学和航海方面的计算；勒内·笛卡尔 (René Descartes ，1596-1650) 引入坐标以将几何简化为代数；艾萨克·牛顿 ( Isaac Newton，1643-1727) 和戈特弗里德·莱布尼茨( Gottfried Leibniz ) 发展了微积分 (1643-1727)。 1646–1716）。 Leonhard Euler （1707-1783）是 18 世纪最著名的数学家，他将这些创新统一到具有标准化术语的单一语料库中，并通过众多定理的发现和证明来完成它们。 95

Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.^[96] In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.^[65]
也许 19 世纪最重要的数学家是德国数学家卡尔·高斯 (Carl Gauss) ，他在代数、分析、微分几何、矩阵论、数论和统计学等领域做出了众多贡献。96 在 20 世纪初，库尔特·哥德尔 (Kurt Gödel)彻底改变了数学通过发表他的不完备定理，该定理部分表明任何一致的公理系统——如果强大到足以描述算术——将包含无法证明的真实命题。 65

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."^[97]
此后，数学得到了极大的扩展，数学和科学之间也进行了富有成效的互动，对双方都有利。直到今天，数学发现仍在继续。据 Mikhail B. Sevryuk 在 2006 年 1 月出版的《美国数学会通报》中指出，“自 1940 年（MR 运行的第一年）以来，数学评论（MR）数据库中收录的论文和书籍数量目前为每年超过 190 万条，超过 75000 个条目被添加到数据库中，这片海洋中的绝大多数作品都包含新的数学定理及其证明。”97

Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas.^[98] More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs,^[99] such as + (plus), × (multiplication), ${\textstyle \int }$ (integral), = (equal), and < (less than).^[100] All these symbols are generally grouped according to specific rules to form expressions and formulas.^[101] Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.
数学符号广泛应用于科学和工程领域，用于以简洁、明确和准确的方式表示复杂的概念和属性。这种表示法由用于表示运算、未指定数字、关系和任何其他数学对象的符号组成，然后将它们组装成表达式和公式。98更准确地说，数字和其他数学对象由称为变量的符号表示，这些符号通常是拉丁语或希腊字母，通常包含下标。运算和关系通常用特定的符号或字形来表示，99例如+ （加）、 × （乘）、 ${\textstyle \int }$ (积分)、 = (等于) 和< (小于)。100 所有这些符号通常按照特定的规则组合起来，形成表达式和公式。 101 通常，表达式和公式不会单独出现，而是包含在句子中当前语言中，表达式充当名词短语的角色，公式充当从句的角色。

Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary.^[102]
数学发展了丰富的术语，涵盖广泛的领域，研究各种抽象、理想化对象的属性以及它们如何相互作用。它基于严格的定义，为通信提供了标准基础。公理或假设是无需证明即可被视为正确的数学陈述。如果一个数学陈述尚未被证明（或反驳），则它被称为猜想。通过一系列采用演绎推理的严格论证，被证明为真的陈述成为定理。主要用于证明另一个定理的专门定理称为引理。构成更普遍发现一部分的已证实实例被称为推论.102

Numerous technical terms used in mathematics are neologisms, such as polynomial and homeomorphism.^[103] Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning.^[104] This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring".
数学中使用的许多技术术语都是新词，例如多项式和同胚。103 其他技术术语是通用语言的单词，其准确含义可能与其常用含义略有不同。例如，在数学中，“ or ”的意思是“一个、另一个或两者”，而在普通语言中，它要么是含糊不清的，要么是“一个或另一个，但不是两者”的意思（在数学中，后者被称为“排他性”）。或者”）。最后，许多数学术语是常用词，但使用时具有完全不同的含义。104这可能会导致句子是正确且真实的数学断言，但对于不具备所需背景的人来说似乎是无意义的。例如，“每个自由模块都是平坦的”和“字段总是一个环”。

Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws.^[105] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.^[106] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.^[107] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.^[108]
大多数科学都使用数学来对现象进行建模，从而可以根据实验定律进行预测。105 数学真理与任何实验的独立性意味着此类预测的准确性仅取决于模型的充分性。106 不准确的预测，这并不是由无效的数学概念引起的，而是意味着需要改变所使用的数学模型。 107 例如，水星的近日点进动只能在爱因斯坦的广义相对论出现后才能解释，广义相对论取代了牛顿的万有引力定律。更好的数学模型.108

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation.^[109] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).^[110] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.^{[111][112][113][114]}
关于数学是否是一门科学，仍然存在着哲学争论。然而，在实践中，数学家通常与科学家归为一类，数学与物理科学有很多共同点。和它们一样，它是可证伪的，这在数学中意味着，如果结果或理论是错误的，可以通过提供反例来证明。与科学类似，理论和结果（定理）通常是从实验中获得的。109 在数学中，实验可能包括对选定示例的计算或对图形或数学对象的其他表示形式的研究（通常是没有物理支持的思维表示形式） 。例如，当被问及他的定理是如何得出的时，高斯曾经回答“durch planmässiges Tattonieren”（通过系统实验）。110然而，一些作者强调，数学与现代科学概念的不同之处在于不依赖经验证据。111112113114

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics.^[115] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.^[116] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece.^[117] The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.^[118]
直到19世纪，西方数学的发展主要是出于技术和科学的需要，纯数学和应用数学之间没有明显的区别。115例如，自然数和算术的引入就是为了满足数学和数学的需要。计数、几何学则受到测量、建筑和天文学的推动。后来，艾萨克·牛顿引入了无穷小微积分，用他的万有引力定律解释了行星的运动。此外，大多数数学家也是科学家，许多科学家也是数学家。 116 然而，古希腊的纯数学传统出现了一个显着的例外。 117 例如，整数分解问题，可以追溯到公元前 300 年的欧几里得，在用于RSA 密码系统之前没有实际应用，现在广泛用于计算机网络的安全。118

In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics.^[115][119] This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.^[120]
19世纪，卡尔·维尔斯特拉斯（Karl Weierstrass）和理查德·戴德金（Richard Dedekind）等数学家越来越将研究重点放在内部问题，即纯数学上。115119这导致数学分裂为纯数学和应用数学，后者通常被认为具有较低的价值在数学纯粹主义者中。然而，两者之间的界限常常是模糊的。120

The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere.^[121][122] Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".^[123][124]
第二次世界大战的后果导致了美国和其他地方应用数学的发展激增。121122 从纯数学的角度来看，许多为应用而开发的理论被发现很有趣，并且显示了许多纯数学的结果在数学之外有应用；反过来，对这些应用的研究可能会给“纯理论”带来新的见解。123124

An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis.^[125] An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high.^[126] For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.^[127]
第一种情况的一个例子是分布理论，由Laurent Schwartz引入，用于验证量子力学中的计算，它立即成为（纯）数学分析的重要工具。 125第二种情况的一个例子是第一种情况的可判定性实数的阶论，这是一个纯数学问题，已被Alfred Tarski证明是正确的，其算法由于计算复杂性太高而无法实现。126 为了获得可以实现的算法，可以求解多项式方程组和不等式，乔治·柯林斯 (George Collins)引入了圆柱代数分解，该分解成为实代数几何中的基本工具 .127

In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.^[128][129] The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".^[24] However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.
如今，纯粹数学和应用数学之间的区别更多地是数学家个人研究目的的问题，而不是将数学划分为广泛的领域。128129 数学学科分类中有一个“普通应用数学”部分，但没有提及“纯数学”。24 然而，这些术语仍然用在一些大学院系的名称中，例如剑桥大学数学系。

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner.^[3] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.^[130] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
数学的不合理有效性是物理学家尤金·维格纳 (Eugene Wigner ) 命名并首先明确指出的一种现象。3 事实上，许多数学理论（甚至是“最纯粹的”）在其最初目标之外都有应用。这些应用可能完全超出了它们最初的数学领域，并且可能涉及数学理论引入时完全未知的物理现象。130数学理论意想不到的应用例子可以在数学的许多领域找到。

A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.^[131] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.^[132]
一个著名的例子是自然数的质因数分解，它的发现早于RSA 密码系统普遍用于安全互联网通信的 2,000 多年。131 第二个历史例子是椭圆理论。古希腊数学家将它们研究为圆锥曲线（即圆锥体与平面的交点）。大约 2000 年后，约翰内斯·开普勒发现行星的轨道是椭圆形。 132

In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four.^[133][134]
19世纪，几何学（纯数学）的内部发展导致了对非欧几里得几何、三维以上空间和流形的定义和研究。此时，这些概念似乎与物理现实完全脱节，但在 20 世纪初，阿尔伯特·爱因斯坦 (Albert Einstein)发展了从根本上使用这些概念的相对论。特别是，狭义相对论的时空是四维的非欧几里得空间，而广义相对论的时空是四维的（弯曲）流形。133134

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon ${\displaystyle \Omega ^{-}.}$ In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.^{[135][136][137]}
数学与物理学之间相互作用的一个引人注目的方面是数学驱动物理学研究。正电子和重子的发现说明了这一点 ${\displaystyle \Omega ^{-}.}$ 在这两种情况下，理论方程都有无法解释的解，这导致人们猜测未知粒子的存在，并寻找这些粒子。在这两种情况下，这些粒子都是几年后通过特定实验发现的。135136137

Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,^[138] and is also considered to be the motivation of major mathematical developments.^[139]
数学和物理学在现代史上相互影响。现代物理学大量使用数学，138，也被认为是数学重大发展的动力。 139

Computing is closely related to mathematics in several ways.^[140] Theoretical computer science is considered to be mathematical in nature.^[141] Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in cryptography and coding theory. Discrete mathematics is useful in many areas of computer science, such as complexity theory, information theory, and graph theory.^[142] In 1998, the Kepler conjecture on sphere packing seemed to also be partially proven by computer.^[143]
计算在几个方面与数学密切相关。 140 理论计算机科学本质上被认为是数学。 141 通信技术应用可能非常古老的数学分支（例如算术），特别是在传输安全、密码学和编码理论。离散数学在计算机科学的许多领域都很有用，例如复杂性理论、信息论和图论。142 1998 年，关于球堆积的开普勒猜想似乎也得到了计算机的部分证明。143

Biology uses probability extensively in fields such as ecology or neurobiology.^[144] Most discussion of probability centers on the concept of evolutionary fitness.^[144] Ecology heavily uses modeling to simulate population dynamics,^[144][145] study ecosystems such as the predator-prey model, measure pollution diffusion,^[146] or to assess climate change.^[147] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.^[148]
生物学在生态学或神经生物学等领域广泛使用概率。144 大多数概率讨论都集中在进化适应性的概念上。144 生态学大量使用建模来模拟种群动态，144145 研究生态系统，例如捕食者-猎物模型、测量污染扩散、 146 或评估气候变化。147 人口动态可以通过耦合微分方程建模，例如Lotka–Volterra 方程.148

Statistical hypothesis testing, is run on data from clinical trials to determine whether a new treatment works.^[149] Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.^[150]
统计假设检验是根据临床试验的数据进行，以确定新疗法是否有效。149 自 20 世纪初以来，化学已使用计算来模拟三维分子。 150

Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes.^[151] Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.^{[152][153][154]}
构造地质学和气候学使用概率模型来预测自然灾害的风险。151 同样，气象学、海洋学和行星学也由于大量使用模型而使用数学。152153154

Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology,^[155] and psychology.^[156]
社会科学中使用的数学领域包括概率/统计学和微分方程。这些用于语言学、经济学、社会学155 和心理学.156

Often the fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. 'economic man').^[157] In this model, the individual seeks to maximize their self-interest,^[157] and always makes optimal choices using perfect information.^[158] This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticise the concept of Homo economicus. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.^[159]
通常，数理经济学的基本假设是理性个人行为者的假设——经济人（字面意思是经济人）。157 在这个模型中，个人寻求自身利益最大化，157 并且总是利用完美信息做出最优选择。158这种经济学的原子论观点使其能够相对容易地将其思维数学化，因为个体计算被转换为数学计算。这种数学模型可以让人们探索经济机制。有些人拒绝或批评经济人的概念。经济学家指出，真实的人信息有限，会做出错误的选择，并且关心公平、利他主义，而不仅仅是个人利益。 159

Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.^[160]
如果没有数学模型，就很难超越统计观察或无法检验的推测。数学建模使经济学家能够创建结构化框架来检验假设并分析复杂的相互作用。模型提供清晰度和精确度，使理论概念能够转化为可量化的预测，并可以根据现实世界的数据进行测试。 160

At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which explains phases of economic growth or crisis.^[161] Towards the end of the 19th century, mathematicians extended their analysis into geopolitics.^[162] Peter Turchin developed cliodynamics since the 1990s.^[163]
20世纪初，出现了用公式表达历史运动的发展。 1922 年，尼古拉·康德拉季耶夫 (Nikolai Kondratiev)发现了约 50 年之久的康德拉季耶夫周期，它解释了经济增长或危机的各个阶段。161 到 19 世纪末，数学家将他们的分析扩展到地缘政治。162 自 20 世纪 90 年代以来，彼得·图尔钦 (Peter Turchin)发展了气候动力学。 163

Mathematization of the social sciences is not without risk. In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.^[164] The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.^[165][166]
社会科学的数学化并非没有风险。在颇具争议的《时尚废话》（Fashionable Nonsense ，1997 年）一书中，索卡尔和布里克蒙特谴责了社会科学中毫无根据或滥用科学术语，特别是数学或物理学术语。 164 复杂系统的研究（失业演变、商业资本、人口演变）人口等）使用数学知识。然而，计算标准（尤其是失业计算标准）或模型的选择可能会引起争议。165166

The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.^[167]
至少从毕达哥拉斯时代起，数学与物质现实之间的联系就引发了哲学争论。古代哲学家柏拉图认为，反映物质现实的抽象本身就具有存在于时空之外的现实。因此，认为数学对象以某种方式抽象地独立存在的哲学观点通常被称为柏拉图主义。独立于他们可能的哲学观点，现代数学家通常可以被认为是柏拉图主义者，因为他们认为和谈论他们的研究对象是真实的对象。 167

Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.^[135]
Armand Borel将这种数学现实观点总结如下，并引用了GH Hardy 、 Charles Hermite 、 Henri Poincaré和 Albert Einstein 来支持他的观点。 135

Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.^[168] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...
一旦我们确信某事物以与我们相同的形式存在于他人的头脑中并且我们可以一起思考并讨论它，那么它就变得客观（而不是“主观”）。168因为语言数学的概念是如此精确，它非常适合定义存在这种共识的概念。在我看来，这足以让我们感受到数学的客观存在和现实......

Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.^[169]
然而，柏拉图主义和同时存在的抽象观点并不能解释数学的不合理有效性。 169

There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.^[170][171] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.^[170] There is not even consensus on whether mathematics is an art or a science.^[171] Some just say, "mathematics is what mathematicians do".^[170] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.^[172]
关于数学的定义或其认识论地位（即它在其他人类活动中的地位）尚未达成普遍共识。170171 许多专业数学家对数学的定义不感兴趣，或者认为它无法定义。170 甚至没有关于数学是一门艺术还是一门科学的共识。171 有些人只是说，“数学就是数学家所做的事情”。170 这是有道理的，因为他们对于什么是数学、什么不是数学有强烈的共识。大多数提出的定义都试图通过数学的研究对象来定义数学。 172

Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.^[173] In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.^[174] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task. For example, in lieu of a definition, Saunders Mac Lane in Mathematics, form and function summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:^[175]
亚里士多德将数学定义为“数量的科学”，这个定义一直流行到18世纪。然而，亚里士多德也指出，仅关注数量可能无法将数学与物理学等科学区分开来。在他看来，抽象和研究数量作为一种与真实实例“在思想上可分离”的属性，使数学与众不同。 173 在 19 世纪，当数学家开始讨论与物理没有明确关系的主题（例如无限集）时， 174随着20世纪初以来出现并不断出现的大量数学新领域，通过这一研究对象来定义数学成为一项不可能完成的任务。例如，桑德斯·麦克莱恩（Saunders Mac Lane）在《数学、形式和函数》中总结了几个数学领域的基础知识，强调了它们的相互联系，并观察到：175

the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.
数学的发展提供了一个由形式规则、概念和系统紧密连接的网络。该网络的节点与人类活动中有用的程序以及科学中出现的问题密切相关。从活动到正式数学系统的过渡是由各种一般见解和想法指导的。

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction.^[176] Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.^[177]
定义数学的另一种方法是使用其方法。因此，只要能够证明定理（其有效性依赖于证明，即纯逻辑演绎的断言），某个研究领域就可以被认定为数学。 176 其他人则认为数学是对公理集合论的研究，因为这项研究现在是许多现代数学的基础学科。177

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules,^[e] without any use of empirical evidence and intuition.^[f][178] Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision, rigorous proofs can require hundreds of pages to express, such as the 255-page Feit–Thompson theorem.^[g] The emergence of computer-assisted proofs has allowed proof lengths to further expand.^[h][179] The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.^[6]
数学推理需要严谨。这意味着定义必须绝对明确，并且证明必须可简化为推理规则的一系列应用，而无需使用任何经验证据和直觉。f178 严格推理并不是数学所特有的，而是数学中的标准严格程度比其他地方高很多。尽管数学很简洁，但严格的证明可能需要数百页来表达，例如255页的Feit-Thompson定理。g计算机辅助证明的出现使得证明长度进一步扩展。h179这种趋势的结果是准经验主义证明的哲学不能被认为是绝对正确的，但具有一定的概率。6

The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation.
数学中严谨的概念可以追溯到古希腊，那里的社会鼓励逻辑演绎推理。然而，这种严格的方法往往会阻碍对新方法的探索，例如无理数和无穷大的概念。在十六世纪，通过符号符号的使用，证明严格证明的方法得到了加强。
In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.^[6]
18 世纪，社会转型导致数学家通过教学谋生，这导致人们对数学的基本概念进行更仔细的思考。这产生了更严格的方法，同时从几何方法过渡到代数方法，然后是算术证明。6

At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.^[6] It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.^[180]
19世纪末，数学基本概念的定义似乎不够准确，无法避免悖论（非欧几里得几何和维尔斯特拉斯函数）和矛盾（罗素悖论）。这是通过将公理与数学理论的必然推理规则相结合来解决的。重新引入古希腊人开创的公理化方法。6 结果是，“严格性”不再是数学中的相关概念，因为证明要么正确，要么错误，而“严格证明”只是一种重复。严格性的特殊概念在证明的社会化方面发挥作用，其中它可能会被其他数学家明显反驳。当一个证据被接受多年甚至几十年后，它就可以被认为是可靠的。 180

Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.^[181]
尽管如此，“严谨”的概念对于向初学者教授什么是数学证明仍然有用。 181

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant.^[182]
数学具有跨越文化界限和时间段的非凡能力。作为一种人类活动，数学实践具有社会性的一面，其中包括教育、职业、认可、普及等。在教育领域，数学是课程的核心部分，也是STEM学科的重要组成部分。职业数学家的突出职业包括数学教师或教授、统计学家、精算师、金融分析师、经济学家、会计师、商品交易员或计算机顾问.182

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.^[183] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE.^[184] The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt.^[185] Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE).^[186] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy.^[187]
考古证据表明，数学教学早在公元前 2000 年就出现在古巴比伦。 183 古代近东地区的抄写数学训练以及希腊罗马世界从公元前 300 年左右开始，都出土了类似的证据。 184 最古老的数学训练著名的数学教科书是莱茵德纸莎草纸，其历史可以追溯到公元 1977 年。公元前 1650 年在埃及。185 由于书籍稀缺，自吠陀时期（约公元前1500 年至约前500 年）以来，古印度的数学教学都是通过记忆的口头传统来传播的。186 在唐朝（618年- 907 CE），进入国家官僚机构的公务员考试采用了数学课程。 187

Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and 17th century. Most mathematical curricula remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was L'Enseignement Mathématique, which began publication in 1899.^[188] The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component—initially for its military applications.^[189] While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.^[190]
黑暗时代之后，欧洲的数学教育由宗教学校作为Quadrivium的一部分提供。正式的教育学教学始于 16 世纪和 17 世纪的耶稣会学校。大多数数学课程一直停留在基础和实用的水平，直到十九世纪它开始在法国和德国蓬勃发展。最古老的数学教学期刊是《L'Enseignement Mathématique》 ，于 1899 年开始出版。188 西方科学技术的进步导致许多民族国家建立了中央集权的教育体系，以数学为核心组成部分——最初是为了军事目的189 虽然课程内容各不相同，但当今几乎所有国家都花大量时间向学生教授数学。 190

During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.^[191] Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.^[192]
在学校期间，数学能力和积极的期望与该领域的职业兴趣密切相关。教师、家长和同伴群体的反馈动机等外在因素可能会影响对数学的兴趣水平。191 一些学习数学的学生可能会对自己在该学科中的表现产生忧虑或恐惧。这被称为数学焦虑症或数学恐惧症，被认为是影响学业成绩的最突出的疾病。数学焦虑可能是由于多种因素造成的，例如父母和老师的态度、社会刻板印象和个人特质。改变教学方法、与家长和老师的互动以及针对个人的定制治疗可以帮助抵消焦虑。 192

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.^[193][194] An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.^[195]
数学定理的有效性仅依赖于其证明的严格性，理论上可以由计算机程序自动完成。这并不意味着数学工作中没有创造力的空间。相反，许多重要的数学结果（定理）是其他数学家未能解决的问题的解决方案，而解决它们的方法的发明可能是解决过程的基本途径。193194一个极端的例子是阿佩里定理：罗杰阿佩里仅提供了证明的想法，几个月后其他三位数学家才给出了正式证明。 195

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles.^[196] This aspect of mathematical activity is emphasized in recreational mathematics.

Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetics.^[197] Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.^[198]
数学家可以找到数学的美学价值。就像美一样，它很难定义，它通常与优雅相关，优雅涉及简单、对称、完整和普遍等品质。 GH Hardy 在《一位数学家的道歉》中表达了这样的信念：美学考虑本身就足以证明纯数学研究的合理性。他还确定了其他标准，如重要性、意外性和必然性，这些都有助于数学美学。197保罗·埃尔多斯通过谈论“书”来表达这种情感，这本书被认为是最美丽的证明的神圣集合。 1998 年出版的《Proofs from THE BOOK》一书受到 Erdős 的启发，是一本特别简洁且具有启发性的数学论证的集合。一些特别优雅的结果的例子包括欧几里得证明有无限多个素数以及调和分析的快速傅里叶变换.198

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.^[199] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science).^[135] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
有些人认为，将数学视为科学就是淡化其在七种传统文科中的艺术性和历史。199 这种观点差异的一种表现方式是在关于数学结果是创造的（如在艺术中）还是发现的哲学辩论中。 （如科学）。135 休闲数学的流行是许多人在解决数学问题中找到乐趣的另一个标志。

Notes that sound well together to a Western ear are sounds whose fundamental frequencies of vibration are in simple ratios. For example, an octave doubles the frequency and a perfect fifth multiplies it by ${\displaystyle {\frac {3}{2}}}$.^[200][201]

Humans, as well as some other animals, find symmetric patterns to be more beautiful.^[202] Mathematically, the symmetries of an object form a group known as the symmetry group.^[203] For example, the group underlying mirror symmetry is the cyclic group of two elements, ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$. A Rorschach test is a figure invariant by this symmetry,^[204] as are butterfly and animal bodies more generally (at least on the surface).^[205] Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.^[206] Fractals possess self-similarity.^[207][208]

Popular mathematics is the act of presenting mathematics without technical terms.^[209] Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract.^[210] However, popular mathematics writing can overcome this by using applications or cultural links.^[211] Despite this, mathematics is rarely the topic of popularization in printed or televised media.

The most prestigious award in mathematics is the Fields Medal,^[212][213] established in 1936 and awarded every four years (except around World War II) to up to four individuals.^[214][215] It is considered the mathematical equivalent of the Nobel Prize.^[215]

Other prestigious mathematics awards include:^[216]

• The Abel Prize, instituted in 2002^[217] and first awarded in 2003^[218]
• The Chern Medal for lifetime achievement, introduced in 2009^[219] and first awarded in 2010^[220]
• The AMS Leroy P. Steele Prize, awarded since 1970^[221]
• The Wolf Prize in Mathematics, also for lifetime achievement,^[222] instituted in 1978^[223]

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.^[224] This list has achieved great celebrity among mathematicians,^[225] and at least thirteen of the problems (depending how some are interpreted) have been solved.^[224]

A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.^[226] To date, only one of these problems, the Poincaré conjecture, has been solved by the Russian mathematician Grigori Perelman.^[227]
2000 年出版了一份新的七个重要问题清单，题为“千年奖问题”。其中只有一个，即黎曼猜想，重复了希尔伯特的一个问题。解决这些问题中的任何一个都会获得 100 万美元的奖励。226 迄今为止，这些问题中只有一个，即庞加莱猜想，已被俄罗斯数学家格里戈里·佩雷尔曼 (Grigori Perelman)解决。227