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0 | "Mathematics"
Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers ( arithmetic and number theory ), [1] formulas and related structures ( algebra ), [2] shapes and the spaces in which they are contained ( geometry ), [1] and quantit... |
1 | "Mathematics"
Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather th... |
2 | "Mathematics"
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 mathematical areas are developed independently from any application (and are therefore called pure mathematics ), but practical a... |
3 | "Mathematics"
[8] Since its beginning, mathematics were essentially divided into geometry, and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra [b] and infinitesimal calculus were introduced as new areas of the subject. Since then, the interaction between ... |
4 | "Mathematics"
The word mathematics comes from Ancient Greek máthēma ( μάθημα ), meaning "that which is learnt," [9] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. [10] Its adjective ... |
5 | "Mathematics"
In Latin, and in 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 has resulted in several mistranslations. For example, Saint Augustine 's... |
6 | "Mathematics"
[12] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math. [13] Before the Renaissance, mathematics was divided into two main areas: arithmetic — regarding the manipulation of numbers, and geometry — regarding the study of shapes. Some types of ... |
7 | "Mathematics"
This division into four main areas — arithmetic, geometry, algebra, calculus [ verification needed ] — endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were often then considered as part of mathematics, but now are considered as belonging to physics. Some su... |
8 | "Mathematics"
Today, the Mathematics Subject Classification contains no less than sixty-four first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic ) and geometry. (However, several other first-level areas have "geometry" in the... |
9 | "Mathematics"
Number theory began with the manipulation of numbers, that is, natural numbers ( N ), {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ). {\displaystyle (\mathbb {Q} ).} Formerly, number theory was called arithmetic, but nowadays ... |
10 | "Mathematics"
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 to this day despite considerable effort. Number theory includes several subareas, including analytic number theory, alge... |
11 | "Mathematics"
A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement 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 accep... |
12 | "Mathematics"
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] Euclidean geometry was developed without change of methods or scope until the 17th century, w... |
13 | "Mathematics"
This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose... |
14 | "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. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are ... |
15 | "Mathematics"
The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term algebra is derived from the Arabic word al-jabr meaning "the reunion for broken parts" [14] that he used for naming one of these methods in the title of ... |
16 | "Mathematics"
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 that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to u... |
17 | "Mathematics"
Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra. (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipul... |
18 | "Mathematics"
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. Calculus, formerly called infinitesimal calculus, was introduced independently and sim... |
19 | "Mathematics"
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: Discrete mathematics, broadly speaking, is the study of finite mat... |
20 | "Mathematics"
The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would impact much of it. [ further explanation needed ] The two subjects of mathematical logic and set theory have both belonged to mathematics since the end of the 19th century. Before ... |
21 | "Mathematics"
Cantor's work offended many mathematicians not only by considering actually infinite sets, but by showing that this implies different sizes of infinity (see Cantor's diagonal argument ) and the existence of mathematical objects that cannot be computed, or even explicitly described (for example, Hamel base... |
22 | "Mathematics"
This became the foundational crisis of mathematics. [15] 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 mu... |
23 | "Mathematics"
This 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 theory that contains the natural numbers, there are th... |
24 | "Mathematics"
Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories. [16] Applied mathematics is the study of m... |
25 | "Mathematics"
Thus, the activity of applied mathematics is vitally connected with research in pure mathematics. [ examples needed ] Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially probability theory. [ definition needed ] Statisticians... |
26 | "Mathematics"
[ clarification needed ] [e] 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 statistic... |
27 | "Mathematics"
[19] Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation an... |
28 | "Mathematics"
As evidenced by tallies found on bone, 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. [21] [22] Evidence for more complex mathematics does not appear until around 3000 BC, when the Babyl... |
29 | "Mathematics"
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.... |
30 | "Mathematics"
[26] His book, Elements, is widely considered the most successful and influential textbook of all time. [27] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. [28] He developed formulas for calculating the surface area and volume of solids of revolution an... |
31 | "Mathematics"
[32] 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. Other notable developments of Indian mathematics include the m... |
32 | "Mathematics"
[33] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Isaac Newton and Gottfried Leibniz in the 17t... |
33 | "Mathematics"
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 Mathemat... |
34 | "Mathematics"
[34] There is no general consensus about the exact definition or epistemological status of mathematics. [35] [36] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. [35] There is not even consensus on whether mathematics is an art or a sci... |
35 | "Mathematics"
[37] In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. [38] To... |
36 | "Mathematics"
Traditionally, these axioms were selected on the grounds of common-sense, but modern axioms typically express formal guarantees for primitive notions, such as simple objects and relations. The validity of a mathematical proof is fundamentally a matter of rigor, and misunderstanding rigor is a notable caus... |
37 | "Mathematics"
Mathematicians refer to this precision of language and logic as "rigor". The rigor expected in mathematics has varied over time: the ancient Greeks expected detailed arguments, but in Isaac Newton 's time, the methods employed were less rigorous (not because of a different conception of mathematics, but b... |
38 | "Mathematics"
This was part of a wider philosophical program known as logicism, which sees mathematics as primarily an extension of logic. Despite mathematics' concision, many proofs require hundreds of pages to express. The emergence of computer-assisted proofs has allowed proof lengths to further expand. Assisted pro... |
39 | "Mathematics"
[g] In addition to special language, contemporary mathematics makes heavy use of special notation. These symbols also contribute to rigor, both by simplifying the expression of mathematical ideas and allowing routine operations that follow consistent rules. Modern notation makes mathematics much more effi... |
40 | "Mathematics"
To a formalist, mathematics is primarily about formal systems of symbols and rules for combining them. From this point-of-view, even axioms are just privileged formulas in an axiomatic system, given without being derived procedurally from other elements in the system. A maximal instance of formalism was D... |
41 | "Mathematics"
Only very exceptional results are accepted as not fitting into one axiomatic system or another. [43] 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 s... |
42 | "Mathematics"
For example, when asked how he came about his theorems, Gauss (one of the greatest mathematicians of the 19th century) once replied "durch planmässiges Tattonieren" (through systematic experimentation). [h] However, some authors emphasize that mathematics differs from the modern notion of science by not r... |
43 | "Mathematics"
String theory, on the other hand, is a proposed framework for unifying much of modern physics that has inspired new techniques and results in mathematics. [49] The German mathematician Carl Friedrich Gauss called mathematics "the Queen of the Sciences", [50] and more recently, Marcus du Sautoy has describ... |
44 | "Mathematics"
[52] Mathematics shows a remarkable tendency to evolve though, and in time, mathematicians often discover surprising applications or links between concepts. One very influential instance of this was the Erlangen program of Felix Klein, which established innovative and profound links between geometry and a... |
45 | "Mathematics"
Several areas of applied mathematics have even merged with practical fields to become disciplines in their own right, such as statistics, operations research, and computer science. Perhaps even more surprising is when ideas flow in the other direction, and even the "purest" mathematics lead to unexpected ... |
46 | "Mathematics"
The ancient philosopher Plato argued this was possible because material reality reflects abstract objects that exist outside time. As a result, the view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. While most mathematicians don't typically concern ... |
47 | "Mathematics"
Qualities like simplicity, symmetry, completeness, and generality are particularly valued in proofs and techniques. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified ... |
48 | "Mathematics"
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts. [56] 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). [57] ... |
49 | "Mathematics"
This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle. [59] In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. However, these programs have motiv... |
50 | "Mathematics"
While the content of courses varies, many countries in the world teach mathematics to students for significant amounts of time. [60] The most prestigious award in mathematics is the Fields Medal, [61] [62] established in 1936 and awarded every four years (except around World War II ) to up to four individ... |
51 | "Mathematics"
[73] 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. [74] To date, only one of these problems, the... |
52 | "Quantity"
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among t... |
53 | "Quantity"
Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: army, fleet, flock, government, company, party, people, mess (military), chorus, crowd, and number; all which are cases of collective nouns. Under the name of magnitude comes what is cont... |
54 | "Quantity"
In mathematics, the concept of quantity is an ancient one extending back to the time of Aristotle and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's ontology, quantity or quantum was classified into two different types, which he characterized as foll... |
55 | "Quantity"
Of these, limited plurality is number, limited length is a line, breadth a surface, depth a solid. In his Elements, Euclid developed the theory of ratios of magnitudes without studying the nature of magnitudes, as Archimedes, but giving the following significant definitions: A magnitude is a part of a magnit... |
56 | "Quantity"
That is, the ratio of magnitudes of any quantity, whether volume, mass, heat and so on, is a number. Following this, Newton then defined number, and the relationship between quantity and number, in the following terms: By number we understand not so much a multitude of unities, as the abstracted ratio of any... |
57 | "Quantity"
The linear continuum represents the prototype of continuous quantitative structure as characterized by Hölder (1901) (translated in Michell & Ernst, 1996). A fundamental feature of any type of quantity is that the relationships of equality or inequality can in principle be stated in comparisons between parti... |
58 | "Quantity"
Another feature is continuity, on which Michell (1999, p. 51) says of length, as a type of quantitative attribute, "what continuity means is that if any arbitrary length, a, is selected as a unit, then for every positive real number, r, there is a length b such that b = r a". A further generalization is give... |
59 | "Quantity"
In formal terms, quantities—their ratios, proportions, order and formal relationships of equality and inequality—are studied by mathematics. The essential part of mathematical quantities consists of having a collection of variables, each assuming a set of values. These can be a set of a single quantity, refe... |
60 | "Quantity"
In mathematics, magnitudes and multitudes are also not only two distinct kinds of quantity but furthermore relatable to each other. Number theory covers the topics of the discrete quantities as numbers: number systems with their kinds and relations. Geometry studies the issues of spatial magnitudes: straight... |
61 | "Quantity"
The theory fits reasonably well elementary or school mathematics but less well the abstract topological and algebraic structures of modern mathematics. [1] Establishing quantitative structure and relationships between different quantities is the cornerstone of modern physical sciences. Physics is fundamental... |
62 | "Quantity"
A distinction has also been made between intensive quantity and extensive quantity as two types of quantitative property, state or relation. The magnitude of an intensive quantity does not depend on the size, or extent, of the object or system of which the quantity is a property, whereas magnitudes of an ext... |
63 | "Quantity"
The quantity is expressed by identifiers, definite and indefinite, and quantifiers, definite and indefinite, as well as by three types of nouns: 1. count unit nouns or countables; 2. mass nouns, uncountables, referring to the indefinite, unidentified amounts; 3. nouns of multitude ( collective nouns ). The w... |
64 | "Quantity"
The set of language quantifiers covers "a few, a great number, many, several (for count names); a bit of, a little, less, a great deal (amount) of, much (for mass names); all, plenty of, a lot of, enough, more, most, some, any, both, each, either, neither, every, no". For the complex case of unidentified amo... |
65 | "Mathematical structure"
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology ). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possi... |
66 | "Mathematical structure"
[1] Mappings between sets which preserve structures (i.e., structures in the domain are mapped to equivalent structures in the codomain ) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topolo... |
67 | "Space"
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. [1] In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuu... |
68 | "Space"
"space"), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later "geometrical conception of place" as "space qua extension" in the Discourse on Place ( Qawl fi al-Makan ) of the 11th-century Arab polymath Alhazen. [2] Many of these classical philosophical questi... |
69 | "Space"
In the 18th century, the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside wo... |
70 | "Space"
According to Albert Einstein 's theory of general relativity, space around gravitational fields deviates from Euclidean space. [4] Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. Galilean and Cartesian theories about space, mat... |
71 | "Space"
[6] As one of the pioneers of modern science, Galileo revised the established Aristotelian and Ptolemaic ideas about a geocentric cosmos. He backed the Copernican theory that the universe was heliocentric, with a stationary sun at the center and the planets—including the Earth—revolving around the sun. If the E... |
72 | "Space"
This view displaced another Aristotelian idea—that all objects gravitated towards their designated natural place-of-belonging. [7] Descartes set out to replace the Aristotelian worldview with a theory about space and motion as determined by natural laws. In other words, he sought a metaphysical foundation or a ... |
73 | "Space"
He is famously known for his "cogito ergo sum" (I think therefore I am), or the idea that we can only be certain of the fact that we can doubt, and therefore think and therefore exist. His theories belong to the rationalist tradition, which attributes knowledge about the world to our ability to think rather tha... |
74 | "Space"
Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together". [10] Unoccupied regions are those that could have objects in them... |
75 | "Space"
Although people in the family are related to one another, the relations do not exist independently of the people. [12] Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes exactly alike except for the location of the material wor... |
76 | "Space"
[13] Newton took space to be more than relations between material objects and based his position on observation and experimentation. For a relationist there can be no real difference between inertial motion, in which the object travels with constant velocity, and non-inertial motion, in which the velocity chang... |
77 | "Space"
After a while, as the bucket continues to spin, the surface of the water becomes concave. If the bucket's spinning is stopped then the surface of the water remains concave as it continues to spin. The concave surface is therefore apparently not the result of relative motion between the bucket and the water. [15... |
78 | "Space"
[16] According to Kant, knowledge about space is synthetic, in that statements about space are not simply true by virtue of the meaning of the words in the statement. In his work, Kant rejected the view that space must be either a substance or relation. Instead he came to the conclusion that space and time are ... |
79 | "Space"
It states that on any plane on which there is a straight line L 1 and a point P not on L 1, there is exactly one straight line L 2 on the plane that passes through the point P and is parallel to the straight line L 1. Until the 19th century, few doubted the truth of the postulate; instead debate centered over w... |
80 | "Space"
In this geometry, an infinite number of parallel lines pass through the point P. Consequently, the sum of angles in a triangle is less than 180° and the ratio of a circle 's circumference to its diameter is greater than pi. In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in... |
81 | "Space"
Carl Friedrich Gauss, a German mathematician, was the first to consider an empirical investigation of the geometrical structure of space. He thought of making a test of the sum of the angles of an enormous stellar triangle, and there are reports that he actually carried out a test, on a small scale, by triangul... |
82 | "Space"
In this world, the temperature is taken to vary in such a way that all objects expand and contract in similar proportions in different places on the sphere. With a suitable falloff in temperature, if the scientists try to use measuring rods to determine the sum of the angles in a triangle, they can be deceived ... |
83 | "Space"
For him, which geometry was used to describe space was a matter of convention. [22] Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed the former would always be used to describe the 'true' geometry of the world. [23] In 1905, Albert Einstein published his special theory of relativity, ... |
84 | "Space"
Moreover, an observer will measure a moving clock to tick more slowly than one that is stationary with respect to them; and objects are measured to be shortened in the direction that they are moving with respect to the observer. Subsequently, Einstein worked on a general theory of relativity, which is a theory ... |
85 | "Space"
Scientists have studied the behaviour of binary pulsars, confirming the predictions of Einstein's theories, and non-Euclidean geometry is usually used to describe spacetime. In modern mathematics spaces are defined as sets with some added structure. They are frequently described as different types of manifolds,... |
86 | "Space"
Space is one of the few fundamental quantities in physics, meaning that it cannot be defined via other quantities because nothing more fundamental is known at the present. On the other hand, it can be related to other fundamental quantities. Thus, similar to other fundamental quantities (like time and mass ), s... |
87 | "Space"
Einstein's discoveries showed that due to relativity of motion our space and time can be mathematically combined into one object– spacetime. It turns out that distances in space or in time separately are not invariant with respect to Lorentz coordinate transformations, but distances in Minkowski space along spa... |
88 | "Space"
Furthermore, in Einstein's general theory of relativity, it is postulated that spacetime is geometrically distorted – curved – near to gravitationally significant masses. [25] One consequence of this postulate, which follows from the equations of general relativity, is the prediction of moving ripples of spacet... |
89 | "Space"
It appears that space was created in the Big Bang, 13.8 billion years ago [28] and has been expanding ever since. The overall shape of space is not known, but space is known to be expanding very rapidly due to the cosmic inflation. The measurement of physical space has long been important. Although earlier soci... |
90 | "Space"
This definition coupled with present definition of the second is based on the special theory of relativity in which the speed of light plays the role of a fundamental constant of nature. Geography is the branch of science concerned with identifying and describing places on Earth, utilizing spatial awareness to ... |
91 | "Space"
While some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership, while still other cultures such as Australian Aboriginals, rather than asserting ownership rights to land, invert the relationship and consider that they are in... |
92 | "Space"
Ownership of space is not restricted to land. Ownership of airspace and of waters is decided internationally. Other forms of ownership have been recently asserted to other spaces—for example to the radio bands of the electromagnetic spectrum or to cyberspace. Public space is a term used to define areas of land ... |
93 | "Space"
When modeling activity or behavior, it is a conceptual tool used to limit extraneous variables such as terrain. Psychologists first began to study the way space is perceived in the middle of the 19th century. Those now concerned with such studies regard it as a distinct branch of psychology. Psychologists analy... |
94 | "Space"
Several space-related phobias have been identified, including agoraphobia (the fear of open spaces), astrophobia (the fear of celestial space) and claustrophobia (the fear of enclosed spaces). The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious infere... |
95 | "Space"
The topic has garnered attention since the 1980s, after the publication of Henri Lefebvre 's The Production of Space. In this book, Lefebvre applies Marxist ideas about the production of commodities and accumulation of capital to discuss space as a social product. His focus is on the multiple and overlapping so... |
96 | "Space"
These advances create relationships across time and space, new markets and groups of wealthy elites in urban centers, all of which annihilate distances and affect our perception of linearity and distance. [31] In his book Thirdspace, Edward Soja describes space and spatiality as an integral and neglected aspect... |
97 | "Space"
Lefebvre's "lived space" [33] and Soja's "thirdspace" are terms that account for the complex ways in which humans understand and navigate place, which "firstspace" and "Secondspace" (Soja's terms for material and imagined spaces respectively) do not fully encompass. Postcolonial theorist Homi Bhabha 's concept ... |
98 | "Calculus"
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals ", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus ... |
99 | "Calculus"
[1] Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. [2] [3] Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, an... |
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数学百科常识向量检索数据库 — ToolElite / wiki_math
本仓库用于服务 ToolElite 的 search 工具，为多轮数学推理提供外部知识检索。 内容为数学百科知识、概念、背景与相关常识。
本仓库包含 corpus、原始数据、构建脚本和预构建 FAISS 索引；检索编码器使用固定版本的 intfloat/e5-base-v2。
Corpus 格式
search_r1_corpus.jsonl 共 573,011 个片段，字段为 id (string), contents (string)。
采用 Search-R1 本地数据库的 JSONL corpus 结构：每行一条记录，id 为按行递增的字符串，
contents 为 "文章标题"\n正文片段。
字段与 ToolElite 本地 corpus 加载器匹配；应保持 corpus 行顺序与向量索引对应。
Search-R1 为格式/流程参考，不是本语料的作者。
原始文件保存在 raw_data/，不与主 corpus 混作训练行；原始处理脚本按原样保存，可能含原机器绝对路径，运行前需指定本机输入/输出路径。
asset-summary.json 与 publication-manifest.json 提供结构摘要、大小和 SHA256 校验。
下载与接入
hf download BitZone910/wiki_math --repo-type dataset --local-dir data/wiki_math
# 在 ToolElite 项目环境中，将 corpus 编码为 FAISS 索引
python -m toolelite.retrieval.general.build_index \
--corpus_path data/wiki_math/search_r1_corpus.jsonl \
--save_dir data/wiki_math/index --faiss_type Flat \
--device cpu --no-faiss_gpu --batch_size 128
检索服务的 general_corpus_path 指向上述 corpus，general_index_path 指向生成的 e5_Flat.index，
并加载与建索引相同的 E5 编码器。两个仓库可以各自建索引；若合并 corpus，需重新编号并重新建索引。
来源、许可与处理
来自本地数学 Wikipedia 抓取快照；25 个原始 JSON 的 link 保留文章 URL，页面历史可用于作者归属。原始下载地址和抓取时间未记录。处理后文本遵循 Wikipedia 的 CC BY-SA 4.0 归属与相同方式共享要求；本文与原始链接一并保留归属。Wikimedia 条款。
清洗、移除标记及按片段切分会改变原文本；详见仓库处理脚本。此数据库不保证所有陈述无误，检索结果需结合推理核实。
预构建 General Search 索引
index/e5_Flat.index 为 ToolElite 原有完整索引，共 573,011 个 768 维向量，类型 IndexFlatIP，使用 inner-product 检索。
索引与本仓库 search_r1_corpus.jsonl 的行顺序绑定，编码器固定为 intfloat/e5-base-v2 的 f52bf8ec8c7124536f0efb74aca902b2995e5bcd。文件 SHA256、大小、核验边界见 index/index-metadata.json。不要替换或重排 corpus 后继续使用该索引。
hf download BitZone910/wiki_math index/e5_Flat.index index/index-metadata.json search_r1_corpus.jsonl --repo-type dataset --local-dir wiki_math
恢复到 ToolElite 的外部资产目录时，将索引放到 assets/retrieval/general_search/index/e5_Flat.index，corpus 放到 assets/retrieval/general_search/corpus/search_r1_corpus.jsonl。索引以原样上传，不重复上传编码器权重。
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