| Mathematics | |
| Article | |
| Talk | |
| Read | |
| View source | |
| View history | |
| Tools | |
| Appearance hide | |
| Birthday mode (Baby Globe) | |
| Disabled | |
| Enabled | |
| Learn more about Birthday mode | |
| Text | |
| Small | |
| Standard | |
| Large | |
| Width | |
| Standard | |
| Wide | |
| Color (beta) | |
| Automatic | |
| Light | |
| Dark | |
| Page semi-protected | |
| From Wikipedia, the free encyclopedia | |
| "Math" and "Maths" redirect here. For other uses, see Mathematics (disambiguation) and Math (disambiguation). | |
| Part of a series on | |
| Mathematics | |
| HistoryIndex | |
| Areas | |
| Number theoryGeometryAlgebraCalculus and AnalysisDiscrete mathematicsLogicProbability and StatisticsDecision theory | |
| Relationship with sciences | |
| PhysicsChemistryGeosciencesComputationBiologyLinguisticsEconomicsPhilosophyEducation | |
| Mathematics Portal | |
| vte | |
| Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved either in response to the needs of empirical sciences or the needs of mathematics itself. There are many areas of mathematics, including 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). | |
| Mathematics involves the description and manipulation of abstract objects that are either abstractions from nature or purely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove the properties of objects through proofs, which consist of a succession of applications of deductive rules to already established results. These results, called theorems, 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] | |
| 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 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 but often find practical applications later.[2][3] | |
| Mathematical written records first appeared in Ancient Egypt and Mesopotamia, but the concept of proof and its associated mathematical rigour began in Ancient Greek mathematics, exemplified in Euclid's Elements.[4] Mathematics was primarily divided into geometry and arithmetic until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus evolved into 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 systematic use 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.[7][8] | |
| Areas of mathematics | |
| Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the study and manipulation of numbers, and geometry, regarding the study of shapes.[9] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[10] | |
| Beginning with the Renaissance, two more areas became predominant. New mathematical notation led to modern algebra which, roughly speaking, begins with the study and manipulation of algebraic expressions. Calculus, consisting of the two subfields differential calculus and integral calculus, originated with geometry but evolved into 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, and calculus[11]—endured until the end of the 19th century. Other areas that were previously studied by mathematicians, such as celestial mechanics and solid mechanics, are now considered as belonging to physics.[12] The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the 17th century.[13] | |
| At the end of the 19th century, the foundational crisis in mathematics and the systematic use of the axiomatic method led to an explosion of new areas of mathematics.[14][6] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[8] 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 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.[7] | |
| Number theory | |
| Main article: Number theory | |
| This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F. | |
| Number theory evolved from 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} ).} Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.[15] The study of numbers arguably dates back to ancient Babylon and probably China, but developed into a distinct discipline in Ancient Greece. Two prominent early number theorists were Euclid and Diophantus of Alexandria.[16] 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.[17] | |
| 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.[18] 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.[19] | |
| Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), Diophantine analysis, and transcendence theory (problem oriented).[7] | |
| Geometry | |
| Main article: Geometry | |
| On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°. | |
| 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.[20] | |
| 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.[21][22] | |
| 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.[b][20] | |
| 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.[23] | |
| 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.[20] | |
| 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.[24][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.[25] | |
| Today's subareas of geometry include:[7] | |
| 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. | |
| Affine geometry, the study of properties relative to parallelism and independent from the concept of length. | |
| Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions. | |
| Manifold theory, the study of shapes that are not necessarily embedded in a larger space. | |
| Riemannian geometry, the study of distance properties in curved spaces. | |
| Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials. | |
| Topology, the study of properties that are kept under continuous deformations. | |
| Algebraic topology, the use in topology of algebraic methods, mainly homological algebra. | |
| Discrete geometry, the study of finite configurations in geometry. | |
| Convex geometry, the study of convex sets, which takes its importance from its applications in optimization. | |
| Complex geometry, the geometry obtained by replacing real numbers with complex numbers. | |
| Algebra | |
| Main article: Algebra | |
| x = (-b ± sqrt(b^2 - 4ac))/2a | |
| The quadratic formula, which concisely expresses the solutions of all quadratic equations | |
| A shuffled 3x3 rubik's cube | |
| The Rubik's Cube group is a concrete application of group theory.[26] | |
| Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.[27][28] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.[29] Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side.[30] 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.[31][32] | |
| 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.[33] Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.[34] | |
| 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.[35] 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,[36] and popularized by Van der Waerden's book Moderne Algebra. | |
| 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:[7] | |
| group theory | |
| field theory | |
| vector spaces, whose study is essentially the same as linear algebra | |
| ring theory | |
| commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry | |
| homological algebra | |
| Lie algebra and Lie group theory | |
| Boolean algebra, which is widely used for the study of the logical structure of computers | |
| The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.[37] 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.[38] | |
| Calculus and analysis | |
| Main articles: Calculus and Mathematical analysis | |
| A Cauchy sequence consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses (from left to right). | |
| Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.[39] It is fundamentally the study of the relationship between variables that depend continuously 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.[40] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.[41] | |
| 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:[7] | |
| Multivariable calculus | |
| Functional analysis, where variables represent varying functions | |
| Integration, measure theory and potential theory, all strongly related with probability theory on a continuum | |
| Ordinary differential equations | |
| Partial differential equations | |
| Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications | |
| Discrete mathematics | |
| Main article: Discrete mathematics | |
| A diagram representing a two-state Markov chain. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state. | |
| Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.[42] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.[c] Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.[43] | |
| The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.[44] 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.[45] | |
| Discrete mathematics includes:[7] | |
| 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. | |
| Graph theory and hypergraphs | |
| Coding theory, including error correcting codes and a part of cryptography | |
| Matroid theory | |
| Discrete geometry | |
| Discrete probability distributions | |
| Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete) | |
| Discrete optimization, including combinatorial optimization, integer programming, constraint programming | |
| Mathematical logic and set theory | |
| Main articles: Mathematical logic and Set theory | |
| A blue and pink circle and their intersection labeled | |
| The Venn diagram is a commonly used method to illustrate the relations between sets. | |
| The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.[46][47] 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.[48] | |
| 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[49] 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.[50] In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour.[51] | |
| This became the foundational crisis of mathematics.[52] 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.[14] 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.[53] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.[54] | |
| 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.[55] 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).[56][57] | |
| 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.[7] 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.[58] | |
| Computational mathematics | |
| Main article: Computational mathematics | |
| Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.[59][60] Part of computational mathematics involves numerical analysis, which is the study of 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.[61] 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.[7] | |
| History | |
| Main article: History of mathematics | |
| Etymology | |
| The word mathematics comes from the Ancient Greek word máthēma (μάθημα), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the Late Middle English period through French and Latin.[62] | |
| Traditionally, 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.[63] | |
| 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.[64] | |
| 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.[65] In English, the noun mathematics takes a singular verb. It is often shortened to maths[66] or, in North America, math.[67] | |
| Ancient | |
| 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.[68][69] | |
| Image of Problem 14 from the Egyptian Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid. | |
| Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[70] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[71]The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[72] Megalithic structures located in Nabta Playa, Upper Egypt featured astronomy, calendar arrangements in alignment with the heliacal rising of Sirius and supported calibration the yearly calendar for the annual Nile flood.[73] Ancient Nubians established a system of geometrics which served as the basis for initial sunclocks. Nubians also exercised a trigonometric methodology comparable to their Egyptian counterparts.[74][75][76][77] | |
| The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC | |
| 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] | |
| 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] | |
| The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD | |
| 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] | |
| Medieval and later | |
| A page from al-Khwarizmi's Al-Jabr | |
| 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] | |
| 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] | |
| Carl Friedrich Gauss | |
| 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.[55] | |
| 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] | |
| Symbolic notation and terminology | |
| Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics | |
| An explanation of the sigma (Σ) summation notation | |
| 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. | |
| 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] | |
| 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". | |
| Relationship with sciences | |
| 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] | |
| 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] | |
| Pure and applied mathematics | |
| Main articles: Applied mathematics and Pure mathematics | |
| Isaac Newton | |
| Gottfried Wilhelm von Leibniz | |
| Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus. | |
| Until the 19th century, there was no clear distinction between pure and applied mathematics as understood today.[115] The distinction between developing mathematics for its own sake or for its applications was rather fluid: natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture, and astronomy, but both subjects quickly stood on their own. Later, Isaac Newton used infinitesimal calculus in part to help explain the movement of the planets and his law of gravitation. Moreover, since antiquity, most mathematicians were also scientists, and many scientists were also mathematicians.[116] Nonetheless, the Western tradition of pure mathematics traces its roots back to 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] | |
| 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] | |
| 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] | |
| 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] | |
| 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".[7] However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge. | |
| Unreasonable effectiveness | |
| 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. | |
| 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 was almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.[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] | |
| 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 | |
| Ω | |
| − | |
| . | |
| {\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] | |
| Specific sciences | |
| Physics | |
| Main article: Relationship between mathematics and physics | |
| Diagram of a pendulum | |
| 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] | |
| Computing | |
| Further information: Theoretical computer science and Computational mathematics | |
| 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] | |
| Statistics and other decision sciences | |
| Main articles: Statistics and Probability theory | |
| Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.[144] | |
| 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 such as, and especially, probability theory. Statisticians generate data with random sampling or randomized experiments.[145] | |
| 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.[146] 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.[147] | |
| Biology and chemistry | |
| Main articles: Mathematical and theoretical biology and Mathematical chemistry | |
| The skin of this giant pufferfish exhibits a Turing pattern, which can be modeled by reaction–diffusion systems. | |
| Biology uses probability extensively in fields such as ecology or neurobiology.[148] Most discussion of probability centers on the concept of evolutionary fitness.[148] Ecology heavily uses modeling to simulate population dynamics,[148][149] study ecosystems such as the predator-prey model, measure pollution diffusion,[150] or to assess climate change.[151] The dynamics of a population can be modeled by coupled differential equations, such as the Lotka–Volterra equations.[152] | |
| Statistical hypothesis testing is run on data from clinical trials to determine whether a new treatment works.[153] Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.[154] | |
| Earth sciences | |
| Main article: Geomathematics | |
| Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes.[155] Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models.[156][157][158] | |
| Social sciences | |
| Further information: Mathematical economics and Historical dynamics | |
| Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, economics, sociology,[159] and psychology.[160] | |
| Supply and demand curves, like this one, are a staple of mathematical economics. | |
| Often the fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus (lit. 'economic man').[161] In this model, the individual seeks to maximize their self-interest,[161] and always makes optimal choices using perfect information.[162] 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 and altruism, not just personal gain.[163] | |
| 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.[164] | |
| 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.[165] Towards the end of the 19th century, mathematicians extended their analysis into geopolitics.[166] Peter Turchin developed cliodynamics in the 1990s.[167] | |
| 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.[168] 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.[169][170] | |
| Philosophy | |
| Main article: Philosophy of mathematics | |
| Reality | |
| 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.[171] | |
| 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] | |
| 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.[172] 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 ... | |
| Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality).[173] | |
| Proposed definitions | |
| Main article: Definitions of mathematics | |
| There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do".[174][175] A common approach is to define mathematics by its object of study.[176][177][178][179] | |
| 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.[180] 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.[181] With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult.[182] 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:[183] | |
| 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. | |
| Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely logical deduction.[d][184][failed verification] | |
| Rigor | |
| See also: Logic | |
| 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][185] 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][186] 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] | |
| 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. 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] | |
| 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.[187] | |
| Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.[188] | |
| Training and practice | |
| Education | |
| Main article: Mathematics education | |
| 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 mathematics teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant.[189] | |
| Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.[190] 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.[191] The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt.[192] 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).[193] 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.[194] | |
| 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.[195] 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.[196] While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.[197] | |
| 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.[198] Some students studying mathematics may develop an apprehension or fear about their performance in the subject. This is known as mathematical anxiety, and is considered the most prominent of the disorders impacting academic performance. Mathematical 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.[199] | |
| Psychology (aesthetic, creativity and intuition) | |
| 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.[200][201] 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.[202] | |
| 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.[203] 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.[204] 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.[205] | |
| Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.[206] 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. | |
| Cultural impact | |
| Artistic expression | |
| Main article: Mathematics and art | |
| 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 | |
| 3 | |
| 2 | |
| {\textstyle {\frac {3}{2}}}.[207][208] | |
| Fractal with a scaling symmetry and a central symmetry | |
| Humans, as well as some other animals, find symmetric patterns to be more beautiful.[209] Mathematically, the symmetries of an object form a group known as the symmetry group.[210] For example, the group underlying mirror symmetry is the cyclic group of two elements, | |
| Z | |
| / | |
| 2 | |
| Z | |
| {\displaystyle \mathbb {Z} /2\mathbb {Z} }. A Rorschach test is a figure invariant by this symmetry,[211] as are butterfly and animal bodies more generally (at least on the surface).[212] 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.[213] Fractals possess self-similarity.[214][215] | |
| Popularization | |
| Main article: Popular mathematics | |
| Popular mathematics is the act of presenting mathematics without technical terms.[216] Presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract.[217] However, popular mathematics writing can overcome this by using applications or cultural links.[218] Despite this, mathematics is rarely the topic of popularization in printed or televised media. | |
| Awards and prize problems | |
| Main category: Mathematics awards | |
| The front side of the Fields Medal with an illustration of the Greek polymath Archimedes | |
| The most prestigious award in mathematics is the Fields Medal,[219][220] established by Canadian John Charles Fields in 1936 and awarded every four years (except around World War II) to up to four individuals.[221][222] It is considered the mathematical equivalent of the Nobel Prize.[222] | |
| Other prestigious mathematics awards include:[223] | |
| The Abel Prize, instituted in 2002[224] and first awarded in 2003[225] | |
| The Chern Medal for lifetime achievement, introduced in 2009[226] and first awarded in 2010[227] | |
| The AMS Leroy P. Steele Prize, awarded since 1970[228] | |
| The Wolf Prize in Mathematics, also for lifetime achievement,[229] instituted in 1978[230] | |
| A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[231] This list has achieved great celebrity among mathematicians,[232] and at least thirteen of the problems (depending how some are interpreted) have been solved.[231] | |
| 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.[233] To date, only one of these problems, the Poincaré conjecture, has been solved, by the Russian mathematician Grigori Perelman.[234] | |
| Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe.[1]: 49–71 [2] Modern science is typically divided into two – or three – major branches:[3] the natural sciences, which study the physical world, and the social sciences, which study individuals and societies.[4][5] While referred to as the formal sciences, the study of logic, mathematics, and theoretical computer science are typically regarded as separate because they rely on deductive reasoning instead of the scientific method as their main methodology.[6][7][8][9] Meanwhile, applied sciences are disciplines that use scientific knowledge for practical purposes, such as engineering and medicine.[10][11][12] | |
| The history of science spans the majority of the historical record, with the earliest identifiable predecessors to modern science dating to the Bronze Age in Egypt and Mesopotamia (c. 3000–1200 BCE). Their contributions to mathematics, astronomy, and medicine entered and shaped the Greek natural philosophy of classical antiquity and later medieval scholarship, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes; while further advances, including the introduction of the Hindu–Arabic numeral system, were made during the Golden Age of India and Islamic Golden Age.[13]: 12 [14]: 1–26 [15][16][13]: 163–192 | |
| The recovery and assimilation of Greek works and Islamic inquiries into Western Europe during the Renaissance revived natural philosophy,[13]: 193–224, 225–253 [17] which was later transformed by the Scientific Revolution that began in the 16th century[18] as new ideas and discoveries departed from previous Greek conceptions and traditions.[13]: 357–368 [14]: 274–322 The scientific method soon played a greater role in the acquisition of knowledge, and in the 19th century, many of the institutional and professional features of science began to take shape,[19][20] along with the changing of "natural philosophy" to "natural science".[21] | |
| New knowledge in science is advanced by research from scientists who are motivated by curiosity about the world and a desire to solve problems.[22][23] Contemporary scientific research is often highly collaborative and is frequently carried out by teams in academic and research institutions,[24] government agencies,[13]: 163–192 and companies.[25] At the same time, many major advances—particularly in fundamental science—have come from individual researchers and are widely recognised through major international awards such as the Nobel Prize. The practical results of scientific work have led to the emergence of science policies that seek to prioritise the responsible development of commercial products, health care, public infrastructure, environmental protection, and defense capabilities. | |
| Etymology | |
| The word science has been used in English since the 14th century (Middle English) in the sense of "the state of knowing". The word was borrowed from the Anglo-Norman language as the suffix -cience, which was borrowed from the Latin word scientia, meaning 'knowledge, awareness, understanding', a noun derivative of sciens meaning 'knowing', itself the present active participle of sciō 'to know'.[26] | |
| There are many hypotheses for science's ultimate word origin. According to Michiel de Vaan, Dutch linguist and Indo-Europeanist, sciō may have its origin in the Proto-Italic language as *skije- or *skijo- meaning 'to know', which may originate from Proto-Indo-European language as *skh1-ie, *skh1-io meaning 'to incise'. The Lexikon der indogermanischen Verben proposed sciō is a back-formation of nescīre, meaning 'to not know, be unfamiliar with', which may derive from Proto-Indo-European *sekH- in Latin secāre, or *skh2- from *sḱʰeh2(i)- meaning 'to cut'.[27] | |
| In the past, science was a synonym for "knowledge" or "study", in keeping with its Latin origin. A person who conducted scientific research was called a "natural philosopher" or "man of science".[28] In 1834, William Whewell introduced the term scientist in a review of Mary Somerville's book On the Connexion of the Physical Sciences,[29] crediting it to "some ingenious gentleman" (possibly himself).[30] | |
| History | |
| Main article: History of science | |
| Early history | |
| Main article: Science in the ancient world | |
| Megaliths from Nabta Playa, constructed by Neolithic populations to coordinate astronomical observations located in Aswan, Upper Egypt.[31] | |
| Clay tablet with markings, three columns for numbers and one for ordinals | |
| The Plimpton 322 tablet by the Babylonians records Pythagorean triples, written c. 1800 BCE | |
| Science has no single origin. Rather, scientific thinking emerged gradually over the course of tens of thousands of years,[32][33] taking different forms around the world, and few details are known about the very earliest developments. Women likely played a central role in prehistoric science,[34] as did religious rituals.[35] Some scholars use the term "protoscience" to label activities in the past that resemble modern science in some but not all features;[36][37][38] however, this label has also been criticised as denigrating,[39] or too suggestive of presentism, thinking about those activities only in relation to modern categories.[40] | |
| Direct evidence for scientific processes becomes clearer with the advent of writing systems in the Bronze Age civilisations of Ancient Egypt and Mesopotamia (c. 3000–1200 BCE), creating the earliest written records in the history of science.[13]: 12–15 [14] Although the words and concepts of "science" and "nature" were not part of the conceptual landscape at the time, the ancient Egyptians and Mesopotamians made contributions that would later find a place in Greek and medieval science: mathematics, astronomy, and medicine.[41][13]: 12 | |
| From the 3rd millennium BCE, the ancient Egyptians developed a non-positional decimal numbering system,[42] solved practical problems using geometry,[43] and developed a calendar.[44] Their healing therapies involved drug treatments and the supernatural, such as prayers, incantations, and rituals.[13]: 9 Ancient Nubians pioneered early antibiotics and established a system of geometrics which served as the basis for initial sunclocks. Nubians also exercised a trigonometric methodology comparable to their Egyptian counterparts.[45][46][47][48] | |
| The ancient Mesopotamians used knowledge about the properties of various natural chemicals for manufacturing pottery, faience, glass, soap, metals, lime plaster, and waterproofing.[49] They studied animal physiology, anatomy, behaviour, and astrology for divinatory purposes.[50] The Mesopotamians had an intense interest in medicine and the earliest medical prescriptions appeared in Sumerian during the Third Dynasty of Ur.[49][51] They seem to have studied scientific subjects which had practical or religious applications and had little interest in satisfying curiosity.[49] | |
| Classical antiquity | |
| Main article: Science in classical antiquity | |
| Framed mosaic of philosophers gathering around and conversing | |
| Plato's Academy mosaic, made between 100 BCE and 79 CE, shows many Greek philosophers and scholars. | |
| In classical antiquity, there is no real ancient analogue of a modern scientist. Instead, well-educated, usually upper-class, and almost universally male individuals performed various investigations into nature whenever they could afford the time.[52] Before the invention or discovery of the concept of phusis or nature by the pre-Socratic philosophers, the same words tend to be used to describe the natural "way" in which a plant grows,[53] and the "way" in which, for example, one tribe worships a particular god. For this reason, it is claimed that these men were the first philosophers in the strict sense and the first to clearly distinguish "nature" and "convention".[54] | |
| The early Greek philosophers of the Milesian school, which was founded by Thales of Miletus and later continued by his successors Anaximander and Anaximenes, were the first to attempt to explain natural phenomena without relying on the supernatural.[55] The Pythagoreans developed a complex number philosophy[56]: 467–468 and contributed significantly to the development of mathematical science.[56]: 465 The theory of atoms was developed by the Greek philosopher Leucippus and his student Democritus.[57][58] Later, Epicurus would develop a full natural cosmology based on atomism, and would adopt a "canon" (ruler, standard) which established physical criteria or standards of scientific truth.[59] The Greek doctor Hippocrates established the tradition of systematic medical science[60][61] and is known as "The Father of Medicine".[62] | |
| A turning point in the history of early philosophical science was Socrates' example of applying philosophy to the study of human matters, including human nature, the nature of political communities, and human knowledge itself. The Socratic method as documented by Plato's dialogues is a dialectic method of hypothesis elimination: better hypotheses are found by steadily identifying and eliminating those that lead to contradictions. The Socratic method searches for general commonly held truths that shape beliefs and scrutinises them for consistency.[63] Socrates criticised the older type of study of physics as too purely speculative and lacking in self-criticism.[64] | |
| In the 4th century BCE, Aristotle created a systematic programme of teleological philosophy.[65] In the 3rd century BCE, Greek astronomer Aristarchus of Samos was the first to propose a heliocentric model of the universe, with the Sun at the centre and all the planets orbiting it.[66] Aristarchus's model was widely rejected because it was believed to violate the laws of physics,[66] while Ptolemy's Almagest, which contains a geocentric description of the Solar System, was accepted through the early Renaissance instead.[67][68] The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus.[69] Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopaedia Natural History.[70][71][72] | |
| Positional notation for representing numbers likely emerged between the 3rd and 5th centuries CE along Indian trade routes. This numeral system made efficient arithmetic operations more accessible and would eventually become standard for mathematics worldwide.[73] | |
| Middle Ages | |
| Main article: History of science § Middle Ages | |
| Picture of a peacock on very old paper | |
| The first page of Vienna Dioscurides depicts a peacock, made in the 6th century. | |
| Due to the collapse of the Western Roman Empire, the 5th century saw an intellectual decline, with knowledge of classical Greek conceptions of the world deteriorating in Western Europe.[13]: 194 Latin encyclopaedists of the period such as Isidore of Seville preserved the majority of general ancient knowledge.[74] In contrast, because the Byzantine Empire resisted attacks from invaders, they were able to preserve and improve prior learning.[13]: 159 John Philoponus, a Byzantine scholar in the 6th century, started to question Aristotle's teaching of physics, introducing the theory of impetus.[13]: 307, 311, 363, 402 His criticism served as an inspiration to medieval scholars and Galileo Galilei, who extensively cited his works ten centuries later.[13]: 307–308 [75] | |
| During late antiquity and the Early Middle Ages, natural phenomena were mainly examined via the Aristotelian approach. The approach includes Aristotle's four causes: material, formal, moving, and final cause.[76] Many Greek classical texts were preserved by the Byzantine Empire and Arabic translations were made by Christians, mainly Nestorians and Miaphysites. Under the Abbasids, these Arabic translations were later improved and developed by Arabic scientists.[14]: 62–67 By the 6th and 7th centuries, the neighbouring Sasanian Empire established the medical Academy of Gondishapur, which was considered by Greek, Syriac, and Persian physicians as the most important medical hub of the ancient world.[77] | |
| Islamic study of Aristotelianism flourished in the House of Wisdom established in the Abbasid capital of Baghdad, Iraq[78] and the flourished[79] until the Mongol invasions in the 13th century. Ibn al-Haytham, better known as Alhazen, used controlled experiments in his optical study.[a][81][82] Avicenna's compilation of The Canon of Medicine, a medical encyclopaedia, is considered to be one of the most important publications in medicine and was used until the 18th century.[83] | |
| By the 11th century most of Europe had become Christian,[13]: 204 and in 1088, the University of Bologna emerged as the first university in Europe.[84] As such, demand for Latin translation of ancient and scientific texts grew,[13]: 204 a major contributor to the Renaissance of the 12th century. Renaissance scholasticism in western Europe flourished, with experiments done by observing, describing, and classifying subjects in nature.[85] In the 13th century, medical teachers and students at Bologna began opening human bodies, leading to the first anatomy textbook based on human dissection by Mondino de Luzzi.[86] | |
| Renaissance | |
| Main articles: Science in the Renaissance and Copernican heliocentrism | |
| Drawing of the heliocentric model as proposed by the Copernicus's De revolutionibus orbium coelestium | |
| New developments in optics played a role in the inception of the Renaissance, both by challenging long-held metaphysical ideas on perception, as well as by contributing to the improvement and development of technology such as the camera obscura and the telescope. At the start of the Renaissance, Roger Bacon, Vitello, and John Peckham each built up a scholastic ontology upon a causal chain beginning with sensation, perception, and finally apperception of the individual and universal forms of Aristotle.[80]: Book I A model of vision later known as perspectivism was exploited and studied by the artists of the Renaissance. This theory uses only three of Aristotle's four causes: formal, material, and final.[87] | |
| In the 16th century, Nicolaus Copernicus formulated a heliostatic model of the Solar System, with the Sun positioned near the center of the Universe,[88][89] motionless, with Earth and the other planets orbiting around it in circular motions,[90] modified by epicycles, and at uniform speeds. The Copernican model challenged the dominant geocentric model of Ptolemy, which had placed Earth at the center of the Universe. 16th-century astronomers believed that Copernicus' elimination of the equant was his chief achievement[91] but his model never displaced Ptolemy's, which only fell out of favor 70 years later after Galileo's telescopic observations of 1610.[92] | |
| Scientific Revolution | |
| Main article: Scientific Revolution | |
| Johannes Kepler and others challenged the notion that the only function of the eye is perception, and shifted the main focus in optics from the eye to the propagation of light.[87][93] Kepler is best known, however, for the discovery of Kepler's laws of planetary motion. Kepler did not reject Aristotelian metaphysics and described his work as a search for the Harmony of the Spheres.[94] Galileo had made significant contributions to astronomy, physics and engineering. However, he became persecuted after Pope Urban VIII sentenced him for writing about the heliocentric model.[95] | |
| The printing press was widely used to publish scholarly arguments, including some that disagreed widely with contemporary ideas of nature.[96] Francis Bacon and René Descartes published philosophical arguments in favour of a new type of non-Aristotelian science. Bacon emphasised the importance of experiment over contemplation, questioned the Aristotelian concepts of formal and final cause, promoted the idea that science should study the laws of nature and the improvement of all human life.[97] Descartes emphasised individual thought and argued that mathematics rather than geometry should be used to study nature.[98] | |
| Age of Enlightenment | |
| Main article: Science in the Age of Enlightenment | |
| Title page of the 1687 first edition of Philosophiæ Naturalis Principia Mathematica by Isaac Newton | |
| At the start of the Age of Enlightenment, Isaac Newton formed the foundation of classical mechanics by his Philosophiæ Naturalis Principia Mathematica greatly influencing future physicists.[99] Gottfried Wilhelm Leibniz incorporated terms from Aristotelian physics, now used in a new non-teleological way. This implied a shift in the view of objects: objects were now considered as having no innate goals. Leibniz assumed that different types of things all work according to the same general laws of nature, with no special formal or final causes.[100] | |
| During this time the declared purpose and value of science became producing wealth and inventions that would improve human lives, in the materialistic sense of having more food, clothing, and other things. In Bacon's words, "the real and legitimate goal of sciences is the endowment of human life with new inventions and riches", and he discouraged scientists from pursuing intangible philosophical or spiritual ideas, which he believed contributed little to human happiness beyond "the fume of subtle, sublime or pleasing [speculation]".[101] | |
| Science during the Enlightenment was dominated by scientific societies and academies,[102] which had largely replaced universities as centres of scientific research and development. Societies and academies were the backbones of the maturation of the scientific profession. Another important development was the popularisation of science among an increasingly literate population.[103] Enlightenment philosophers turned to a few of their scientific predecessors – Galileo, Kepler, Boyle, and Newton principally – as the guides to every physical and social field of the day.[104][105] | |
| The 18th century saw significant advancements in the practice of medicine[106] and physics;[107] the development of biological taxonomy by Carl Linnaeus;[108] a new understanding of magnetism and electricity;[109] and the maturation of chemistry as a discipline.[110] Ideas on human nature, society, and economics evolved during the Enlightenment. Hume and other Scottish Enlightenment thinkers developed A Treatise of Human Nature, which was expressed historically in works by authors including James Burnett, Adam Ferguson, John Millar and William Robertson, all of whom merged a scientific study of how humans behaved in ancient and primitive cultures with a strong awareness of the determining forces of modernity.[111] Modern sociology largely originated from this movement.[112] In 1776, Adam Smith published The Wealth of Nations, which is often considered the first work on modern economics.[113] | |
| 19th century | |
| Main article: 19th century in science | |
| Sketch of a map with captions | |
| The first diagram of an evolutionary tree made by Charles Darwin in 1837 | |
| During the 19th century, many distinguishing characteristics of contemporary modern science began to take shape. These included the transformation of the life and physical sciences; the frequent use of precision instruments; the emergence of terms such as "biologist", "physicist", and "scientist"; an increased professionalisation of those studying nature; scientists gaining cultural authority over many dimensions of society; the industrialisation of numerous countries; the thriving of popular science writings; and the emergence of science journals.[114] During the late 19th century, psychology emerged as a separate discipline from philosophy when Wilhelm Wundt founded the first laboratory for psychological research in 1879.[115] | |
| During the mid-19th century Charles Darwin and Alfred Russel Wallace independently proposed the theory of evolution by natural selection in 1858, which explained how different plants and animals originated and evolved. Their theory was set out in detail in Darwin's book On the Origin of Species, published in 1859.[116] Separately, Gregor Mendel presented his paper, "Experiments on Plant Hybridisation" in 1865,[117] which outlined the principles of biological inheritance, serving as the basis for modern genetics.[118] | |
| Early in the 19th century John Dalton suggested the modern atomic theory, based on Democritus's original idea of indivisible particles called atoms.[119] The laws of conservation of energy, conservation of momentum and conservation of mass suggested a highly stable universe where there could be little loss of resources. However, with the advent of the steam engine and the Industrial Revolution there was an increased understanding that not all forms of energy have the same energy qualities, the ease of conversion to useful work or to another form of energy.[120] This realisation led to the development of the laws of thermodynamics, in which the free energy of the universe is seen as constantly declining: the entropy of a closed universe increases over time.[b] | |
| The electromagnetic theory was established in the 19th century by the works of Hans Christian Ørsted, André-Marie Ampère, Michael Faraday, James Clerk Maxwell, Oliver Heaviside, and Heinrich Hertz. The new theory raised questions that could not easily be answered using Newton's framework. The discovery of X-rays inspired the discovery of radioactivity by Henri Becquerel and Marie Curie in 1896,[123] Marie Curie then became the first person to win two Nobel Prizes.[124] In the next year came the discovery of the first subatomic particle, the electron.[125] | |
| 20th century | |
| Main article: 20th century in science | |
| Graph showing lower ozone concentration at the South Pole | |
| A computer graph of the ozone hole made in 1987 using data from a space telescope | |
| In the first half of the century the development of antibiotics and artificial fertilisers improved human living standards globally.[126][127] Harmful environmental issues such as ozone depletion, ocean acidification, eutrophication, and climate change came to the public's attention and caused the onset of environmental studies.[128] | |
| During this period scientific experimentation became increasingly larger in scale and funding.[129] The extensive technological innovation stimulated by World War I, World War II, and the Cold War led to competitions between global powers, such as the Space Race and nuclear arms race.[130][131] Substantial international collaborations were also made, despite armed conflicts.[132] | |
| In the late 20th century active recruitment of women and elimination of sex discrimination greatly increased the number of women scientists, but large gender disparities remained in some fields.[133] The discovery of the cosmic microwave background in 1964[134] led to a rejection of the steady-state model of the universe in favour of the Big Bang theory of Georges Lemaître.[135] | |
| The century saw fundamental changes within science disciplines. Evolution became a unified theory in the early 20th century when the modern synthesis reconciled Darwinian evolution with classical genetics.[136] Albert Einstein's theory of relativity and the development of quantum mechanics complement classical mechanics to describe physics in extreme length, time and gravity.[137][138] Widespread use of integrated circuits in the last quarter of the 20th century combined with communications satellites led to a revolution in information technology and the rise of the global internet and mobile computing, including smartphones. The need for mass systematisation of long, intertwined causal chains and large amounts of data led to the rise of the fields of systems theory and computer-assisted scientific modelling.[139] | |
| 21st century | |
| Main article: 21st century § Science and technology | |
| The Human Genome Project was completed in 2003 by identifying and mapping all of the genes of the human genome.[140] The first induced pluripotent human stem cells were made in 2006, allowing adult cells to be transformed into stem cells and turn into any cell type found in the body.[141] With the affirmation of the Higgs boson discovery in 2013, the last particle predicted by the Standard Model of particle physics was found.[142] In 2015, gravitational waves, predicted by general relativity a century before, were first observed.[143][144] In 2019, the international collaboration Event Horizon Telescope presented the first direct image of a black hole's accretion disc.[145] | |
| Branches | |
| Main article: Branches of science | |
| Modern science is commonly divided into three major branches: natural science, social science, and formal science.[3] Each of these branches comprises various specialised yet overlapping scientific disciplines that often possess their own nomenclature and expertise.[146] Both natural and social sciences are empirical sciences,[147] as their knowledge is based on empirical observations and is capable of being tested for its validity by other researchers working under the same conditions.[148]: 3–26 | |
| Natural | |
| Natural science is the study of the physical world. It can be divided into two main branches: life science and physical science. These two branches may be further divided into more specialised disciplines. For example, physical science can be subdivided into physics, chemistry, astronomy, and earth science. Modern natural science is the successor to the natural philosophy that began in Ancient Greece. Galileo, Descartes, Bacon, and Newton debated the benefits of using approaches that were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, remain necessary in natural science.[149] Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and other biotic beings.[150] Today, "natural history" suggests observational descriptions aimed at popular audiences.[151] | |
| Social | |
| Two curve crossing over at a point, forming a X shape | |
| Supply and demand curve in economics, crossing over at the optimal equilibrium | |
| Social science is the study of human behaviour and the functioning of societies.[4][5] It has many disciplines that include, but are not limited to anthropology, economics, history, human geography, political science, psychology, and sociology.[4] In the social sciences, there are many competing theoretical perspectives, many of which are extended through competing research programmes such as the functionalists, conflict theorists, and interactionists in sociology.[4] Due to the limitations of conducting controlled experiments involving large groups of individuals or complex situations, social scientists may adopt other research methods such as the historical method, case studies, and cross-cultural studies. Moreover, if quantitative information is available, social scientists may rely on statistical approaches to better understand social relationships and processes.[4] | |
| Formal | |
| Formal science is an area of study that generates knowledge using formal systems.[152][153][154] A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules.[155] It includes mathematics,[156][157] systems theory, and theoretical computer science. The formal sciences share similarities with the other two branches by relying on objective, careful, and systematic study of an area of knowledge. They are, however, different from the empirical sciences as they rely exclusively on deductive reasoning, without the need for empirical evidence, to verify their abstract concepts.[8][158][148] The formal sciences are therefore a priori disciplines and because of this, there is disagreement on whether they constitute a science.[6][159] Nevertheless, the formal sciences play an important role in the empirical sciences. Calculus, for example, was initially invented to understand motion in physics.[160] Natural and social sciences that rely heavily on mathematical applications include mathematical physics,[161] chemistry,[162] biology,[163] finance,[164] and economics.[165] | |
| Applied | |
| Applied science is the use of the scientific method and knowledge to attain practical goals and includes a broad range of disciplines such as engineering and medicine.[166][12] Engineering is the use of scientific principles to invent, design and build machines, structures and technologies.[167] Science may contribute to the development of new technologies.[168] Medicine is the practice of caring for patients by maintaining and restoring health through the prevention, diagnosis, and treatment of injury or disease.[169][170] | |
| Basic | |
| The applied sciences are often contrasted with the basic sciences, which are focused on advancing scientific theories and laws that explain and predict events in the natural world.[171][172] | |
| Blue skies | |
| Blue skies research, also called blue sky science, is scientific research in domains where "real-world" applications are not immediately apparent. It has been defined as "research without a clear goal"[173] and "curiosity-driven science". Proponents of this mode of science argue that unanticipated scientific breakthroughs are sometimes more valuable than the outcomes of agenda-driven research, heralding advances in genetics and stem cell biology as examples of unforeseen benefits of research that was originally seen as purely theoretical in scope. Because of the inherently uncertain return on investment, blue-sky projects are sometimes politically and commercially unpopular and tend to lose funding to research perceived as being more reliably profitable or practical.[174] | |
| Computational | |
| Computational science applies computer simulations to science, enabling a better understanding of scientific problems than formal mathematics alone can achieve. The use of machine learning and artificial intelligence is becoming a central feature of computational contributions to science, for example in agent-based computational economics, random forests, topic modelling and various forms of prediction. However, machines alone rarely advance knowledge as they require human guidance and capacity to reason; and they can introduce bias against certain social groups or sometimes underperform against humans.[175][176] | |
| Interdisciplinary | |
| Interdisciplinary science involves the combination of two or more disciplines into one,[177] such as bioinformatics, a combination of biology and computer science[178] or cognitive sciences. The concept has existed since the ancient Greek period and it became popular again in the 20th century.[179] | |
| Research | |
| Scientific research can be labelled as either basic or applied research. Basic research is the search for knowledge and applied research is the search for solutions to practical problems using this knowledge. Most understanding comes from basic research, though sometimes applied research targets specific practical problems. This leads to technological advances that were not previously imaginable.[180] | |
| Scientific method | |
| 6 steps of the scientific method in a loop | |
| A diagram variant of scientific method represented as an ongoing process | |
| Scientific research involves using the scientific method, which seeks to objectively explain the events of nature in a reproducible way.[181] Scientists usually take for granted a set of basic assumptions that are needed to justify the scientific method: there is an objective reality shared by all rational observers; this objective reality is governed by natural laws; these laws were discovered by means of systematic observation and experimentation.[2] Mathematics is essential in the formation of hypotheses, theories, and laws, because it is used extensively in quantitative modelling, observing, and collecting measurements.[148]: 3–26 Statistics is used to summarise and analyse data, which allows scientists to assess the reliability of experimental results.[182] | |
| In the scientific method an explanatory thought experiment or hypothesis is put forward as an explanation using parsimony principles and is expected to seek consilience – fitting with other accepted facts related to an observation or scientific question.[183] This tentative explanation is used to make falsifiable predictions, which are typically posted before being tested by experimentation. Disproof of a prediction is evidence of progress.[181]: 4–5 [184] Experimentation is especially important in science to help establish causal relationships to avoid the correlation fallacy, though in some sciences such as astronomy or geology, a predicted observation might be more appropriate.[185] | |
| When a hypothesis proves unsatisfactory it is modified or discarded. If the hypothesis survives testing, it may become adopted into the framework of a scientific theory, a validly reasoned, self-consistent model or framework for describing the behaviour of certain natural events. A theory typically describes the behaviour of much broader sets of observations than a hypothesis; commonly, a large number of hypotheses can be logically bound together by a single theory. Thus, a theory is a hypothesis explaining various other hypotheses. In that vein, theories are formulated according to most of the same scientific principles as hypotheses. Scientists may generate a model, an attempt to describe or depict an observation in terms of a logical, physical or mathematical representation, and to generate new hypotheses that can be tested by experimentation.[186] | |
| While performing experiments to test hypotheses, scientists may have a preference for one outcome over another.[187][188] Eliminating the bias can be achieved through transparency, careful experimental design, and a thorough peer review process of the experimental results and conclusions.[189][190] After the results of an experiment are announced or published, it is normal practice for independent researchers to double-check how the research was performed, and to follow up by performing similar experiments to determine how dependable the results might be.[191] Taken in its entirety, the scientific method allows for highly creative problem solving while minimising the effects of subjective and confirmation bias.[192] Intersubjective verifiability, the ability to reach a consensus and reproduce results, is fundamental to the creation of all scientific knowledge.[193] | |
| Literature | |
| Main articles: Scientific literature and Lists of important publications in science | |
| Decorated "NATURE" as title, with scientific text below | |
| Cover of the first issue of Nature, 4 November 1869 | |
| Scientific research is published in a range of literature.[194] Scientific journals communicate and document the results of research carried out in universities and various other research institutions, serving as an archival record of science. The first scientific journals, Journal des sçavans followed by Philosophical Transactions, began publication in 1665. Since that time the total number of active periodicals has steadily increased. In 1981, one estimate for the number of scientific and technical journals in publication was 11,500.[195] | |
| Most scientific journals cover a single scientific field and publish the research within that field; the research is normally expressed in the form of a scientific paper. Science has become so pervasive in modern societies that it is considered necessary to communicate the achievements, news, and ambitions of scientists to a wider population.[196] | |
| Philosophy | |
| Depiction of epicycles, where a planet orbit is going around in a bigger orbit | |
| For Kuhn, the addition of epicycles in Ptolemaic astronomy was "normal science" within a paradigm, whereas the Copernican Revolution was a paradigm shift. | |
| There are different schools of thought in the philosophy of science. The most popular position is empiricism, which holds that knowledge is created by a process involving observation; scientific theories generalise observations.[197]: 39–56 Empiricism generally encompasses inductivism, a position that explains how general theories can be made from the finite amount of empirical evidence available. Many versions of empiricism exist, with the predominant ones being Bayesianism and the hypothetico-deductive method.[197]: 219–232 | |
| Empiricism has stood in contrast to rationalism, the position originally associated with Descartes, which holds that knowledge is created by the human intellect, not by observation.[197]: 19–38 Critical rationalism is a contrasting 20th-century approach to science, first defined by Austrian-British philosopher Karl Popper. Popper rejected the way that empiricism describes the connection between theory and observation. He claimed that theories are not generated by observation, but that observation is made in the light of theories, and that the only way theory A can be affected by observation is after theory A were to conflict with observation, but theory B were to survive the observation.[197]: 57–74 Popper proposed replacing verifiability with falsifiability as the landmark of scientific theories, replacing induction with falsification as the empirical method.[197]: 102–121 Popper further claimed that there is actually only one universal method, not specific to science: the negative method of criticism, trial and error,[197]: 102–121 covering all products of the human mind, including science, mathematics, philosophy, and art.[198] | |
| Another approach, instrumentalism, emphasises the utility of theories as instruments for explaining and predicting phenomena. It views scientific theories as black boxes, with only their input (initial conditions) and output (predictions) being relevant. Consequences, theoretical entities, and logical structure are claimed to be things that should be ignored.[199] Close to instrumentalism is constructive empiricism, according to which the main criterion for the success of a scientific theory is whether what it says about observable entities is true.[200] | |
| Thomas Kuhn argued that the process of observation and evaluation takes place within a paradigm, a logically consistent "portrait" of the world that is consistent with observations made from its framing. He characterised normal science as the process of observation and "puzzle solving", which takes place within a paradigm, whereas revolutionary science occurs when one paradigm overtakes another in a paradigm shift.[201] Each paradigm has its own distinct questions, aims, and interpretations. The choice between paradigms involves setting two or more "portraits" against the world and deciding which likeness is most promising. A paradigm shift occurs when a significant number of observational anomalies arise in the old paradigm and a new paradigm makes sense of them. That is, the choice of a new paradigm is based on observations, even though those observations are made against the background of the old paradigm. For Kuhn, acceptance or rejection of a paradigm is a social process as much as a logical process. Kuhn's position, however, is not one of relativism.[202] | |
| Another approach often cited in debates of scientific scepticism against controversial movements like "creation science" is methodological naturalism. Naturalists maintain that a difference should be made between natural and supernatural, and science should be restricted to natural explanations.[197]: 149–162 Methodological naturalism maintains that science requires strict adherence to empirical study and independent verification.[203] | |
| Community | |
| The scientific community is a network of interacting scientists who conduct scientific research. The community consists of smaller groups working in scientific fields. By having peer review, through discussion and debate within journals and conferences, scientists maintain the quality of research methodology and objectivity when interpreting results.[204] | |
| Scientists | |
| Portrait of a middle-aged woman | |
| Marie Curie was the first person to be awarded two Nobel Prizes: Physics in 1903 and Chemistry in 1911.[124] | |
| Scientists are individuals who conduct scientific research to advance knowledge in an area of interest.[205][206] Scientists may exhibit a strong curiosity about reality and a desire to apply scientific knowledge for the benefit of public health, nations, the environment, or industries; other motivations include recognition by peers and prestige.[207] In modern times, many scientists study within specific areas of science in academic institutions, often obtaining advanced degrees in the process.[208] Many scientists pursue careers in various fields such as academia, industry, government, and nonprofit organisations.[209][210][207] | |
| Science has historically been a male-dominated field, with notable exceptions. Women have faced considerable discrimination in science, much as they have in other areas of male-dominated societies. For example, women were frequently passed over for job opportunities and denied credit for their work.[211] The achievements of women in science have been attributed to the defiance of their traditional role as labourers within the domestic sphere.[212] | |
| Learned societies | |
| Scientists at the 200th anniversary of the Prussian Academy of Sciences, 1900 | |
| Learned societies for the communication and promotion of scientific thought and experimentation have existed since the Renaissance.[213] Many scientists belong to a learned society that promotes their respective scientific discipline, profession, or group of related disciplines.[214] Membership may either be open to all, require possession of scientific credentials, or conferred by election.[215] Most scientific societies are nonprofit organisations,[216] and many are professional associations. Their activities typically include holding regular conferences for the presentation and discussion of new research results and publishing or sponsoring academic journals in their discipline. Some societies act as professional bodies, regulating the activities of their members in the public interest, or the collective interest of the membership. | |
| The professionalisation of science, begun in the 19th century, was partly enabled by the creation of national distinguished academies of sciences such as the Italian Accademia dei Lincei in 1603,[217] the British Royal Society in 1660,[218] the French Academy of Sciences in 1666,[219] the American National Academy of Sciences in 1863,[220] the German Kaiser Wilhelm Society in 1911,[221] and the Chinese Academy of Sciences in 1949.[222] International scientific organisations, such as the International Science Council, are devoted to international cooperation for science advancement.[223] | |
| Awards | |
| Science awards are usually given to individuals or organisations that have made significant contributions to a discipline. They are often given by prestigious institutions; thus, it is considered a great honour for a scientist receiving them. Since the early Renaissance, scientists have often been awarded medals, money, and titles. The Nobel Prize, a widely regarded prestigious award, is awarded annually to those who have achieved scientific advances in the fields of medicine, physics, and chemistry.[224] | |
| Society | |
| "Science and society" redirects here; not to be confused with Science & Society or Sociology of scientific knowledge. | |
| Funding and policies | |
| see caption | |
| Budget of NASA as percentage of United States federal budget, peaking at 4.4% in 1966 and slowly declining since | |
| Funding of science is often through a competitive process in which potential research projects are evaluated and only the most promising receive funding. Such processes, which are run by government, corporations, or foundations, allocate scarce funds. Total research funding in most developed countries is between 1.5% and 3% of GDP.[225] In the OECD, around two-thirds of research and development in scientific and technical fields is carried out by industry, and 20% and 10%, respectively, by universities and government. The government funding proportion in certain fields is higher, and it dominates research in social science and the humanities. In less developed nations, the government provides the bulk of the funds for their basic scientific research.[226] | |
| Many governments have dedicated agencies to support scientific research, such as the National Science Foundation in the United States,[227] the National Scientific and Technical Research Council in Argentina,[228] Commonwealth Scientific and Industrial Research Organisation in Australia,[229] National Centre for Scientific Research in France,[230] the Max Planck Society in Germany,[231] and National Research Council in Spain.[232] In commercial research and development, all but the most research-orientated corporations focus more heavily on near-term commercialisation possibilities than research driven by curiosity.[233] | |
| Science policy is concerned with policies that affect the conduct of the scientific enterprise, including research funding, often in pursuance of other national policy goals such as technological innovation to promote commercial product development, weapons development, health care, and environmental monitoring. Science policy sometimes refers to the act of applying scientific knowledge and consensus to the development of public policies. In accordance with public policy being concerned about the well-being of its citizens, science policy's goal is to consider how science and technology can best serve the public.[234] Public policy can directly affect the funding of capital equipment and intellectual infrastructure for industrial research by providing tax incentives to those organisations that fund research.[196] | |
| Education and awareness | |
| Main articles: Public awareness of science and Science journalism | |
| Dinosaur exhibit at the Houston Museum of Natural Science | |
| Science education for the general public is embedded in the school curriculum, and is supplemented by online pedagogical content (for example, YouTube and Khan Academy), museums, and science magazines and blogs. Major organisations of scientists such as the American Association for the Advancement of Science (AAAS) consider the sciences to be a part of the liberal arts traditions of learning, along with philosophy and history.[235] Scientific literacy is chiefly concerned with an understanding of the scientific method, units and methods of measurement, empiricism, a basic understanding of statistics (correlations, qualitative versus quantitative observations, aggregate statistics), and a basic understanding of core scientific fields such as physics, chemistry, biology, ecology, geology, and computation. As a student advances into higher stages of formal education, the curriculum becomes more in depth. Traditional subjects usually included in the curriculum are natural and formal sciences, although recent movements include social and applied science as well.[236] | |
| The mass media face pressures that can prevent them from accurately depicting competing scientific claims in terms of their credibility within the scientific community as a whole. Determining how much weight to give different sides in a scientific debate may require considerable expertise regarding the matter.[237] Few journalists have real scientific knowledge, and even beat reporters who are knowledgeable about certain scientific issues may be ignorant about other scientific issues that they are suddenly asked to cover.[238][239] | |
| Science magazines such as New Scientist, Science & Vie, and Scientific American cater to the needs of a much wider readership and provide a non-technical summary of popular areas of research, including notable discoveries and advances in certain fields of research.[240] The science fiction genre, primarily speculative fiction, can transmit the ideas and methods of science to the general public.[241] Recent efforts to intensify or develop links between science and non-scientific disciplines, such as literature or poetry, include the Creative Writing Science resource developed through the Royal Literary Fund.[242] | |
| Anti-science attitudes | |
| Main article: Antiscience | |
| While the scientific method is broadly accepted in the scientific community, some fractions of society reject certain scientific positions or are sceptical about science. Examples are the common notion that COVID-19 is not a major health threat to the US (held by 39% of Americans in August 2021)[243] or the belief that climate change is not a major threat to the US (also held by 40% of Americans, in late 2019 and early 2020).[244] Psychologists have pointed to several factors driving rejection of scientific results:[245] | |
| Scientific authorities are sometimes seen as inexpert, untrustworthy, or biased. | |
| Some marginalised social groups hold anti-science attitudes, in part because these groups have often been exploited in unethical experiments.[246] | |
| Messages from scientists may contradict deeply held existing beliefs or morals. | |
| Anti-science attitudes often seem to be caused by fear of rejection in social groups. For instance, climate change is perceived as a threat by only 22% of Americans on the right side of the political spectrum, but by 85% on the left.[247] That is, if someone on the left would not consider climate change as a threat, this person may face contempt and be rejected in that social group. In fact, people may rather deny a scientifically accepted fact than lose or jeopardise their social status.[248] | |
| Politics | |
| See also: Politicization of science | |
| Result in bar graph of two questions ("Is global warming occurring?" and "Are oil/gas companies responsible?"), showing large discrepancies between American Democrats and Republicans | |
| Public opinion on global warming in the United States by political party[249] | |
| Attitudes towards science are often determined by political opinions and goals. Government, business and advocacy groups have been known to use legal and economic pressure to influence scientific researchers. Many factors can act as facets of the politicisation of science such as anti-intellectualism, perceived threats to religious beliefs, and fear for business interests.[250] Politicisation of science is usually accomplished when scientific information is presented in a way that emphasises the uncertainty associated with the scientific evidence.[251] Tactics such as shifting conversation, failing to acknowledge facts, and capitalising on doubt of scientific consensus have been used to gain more attention for views that have been undermined by scientific evidence.[252] Examples of issues that have involved the politicisation of science include the global warming controversy, health effects of pesticides, and health effects of tobacco.[252][253] | |
| Challenges | |
| See also: Criticism of science and Academic bias | |
| The replication crisis is an ongoing methodological crisis that affects parts of the social and life sciences. In subsequent investigations, the results of many scientific studies have been proven to be unrepeatable.[254] The crisis has long-standing roots; the phrase was coined in the early 2010s[255] as part of a growing awareness of the problem. The replication crisis represents an important body of research in metascience, which aims to improve the quality of all scientific research while reducing waste.[256] | |
| The term scientific misconduct refers to situations such as where researchers have intentionally misrepresented their published data or have purposely given credit for a discovery to the wrong person.[257] An area of study or speculation that masquerades as science in an attempt to claim legitimacy that it would not otherwise be able to achieve is sometimes referred to as pseudoscience, fringe science, or junk science.[258][259] Physicist Richard Feynman coined the term "cargo cult science" for cases in which researchers believe, and at a glance, look like they are doing science but lack the honesty to allow their results to be rigorously evaluated.[260] Various types of commercial advertising, ranging from hype to fraud, may fall into these categories. Science has been described as "the most important tool" for separating valid claims from invalid ones.[261] Sometimes, research can be well-intended but is incorrect, obsolete, incomplete, or over-simplified expositions of scientific ideas. | |
| There can also be an element of political bias or ideological bias in science. Scientists in some countries were found to have a bias in political party preferences compared to the general population.[262] | |
| Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. | |
| Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work."[1] Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like | |
| ∧ | |
| {\displaystyle \land } (and) or | |
| → | |
| {\displaystyle \to } (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. | |
| Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have the strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens.[2] Abductive arguments are inferences to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer.[3] Arguments that fall short of the standards of correct reasoning often embody fallacies. Systems of logic are theoretical frameworks for assessing the correctness of arguments. | |
| Logic has been studied since antiquity. Early approaches include Aristotelian logic, Stoic logic, Nyaya, and Mohism. Aristotelian logic focuses on reasoning in the form of syllogisms. It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege. Today, the most commonly used system is classical logic. It consists of propositional logic and first-order logic. Propositional logic only considers logical relations between full propositions. First-order logic also takes the internal parts of propositions into account, like predicates and quantifiers. Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics, ethics, and epistemology. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. | |
| Definition | |
| The word "logic" originates from the Greek word logos, which has a variety of translations, such as reason, discourse, or language.[4] Logic is traditionally defined as the study of the laws of thought or correct reasoning,[5] and is usually understood in terms of inferences or arguments. Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences.[6] An argument is a set of premises together with a conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion.[7] These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments.[8] Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic.[9] | |
| Formal logic | |
| Further information: Formal system | |
| Formal logic (also known as symbolic logic) is widely used in mathematical logic. It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content.[10] | |
| Formal logic is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false.[11] For valid arguments, the logical structure that leads from the premises to the conclusion follows a pattern called a rule of inference.[12] For example, modus ponens is a rule of inference according to which all arguments of the form "(1) p, (2) if p then q, (3) therefore q" are valid, independent of what the terms p and q stand for.[13] In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths.[14] A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like the claim "either it is raining, or it is not".[15] These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q" is a logical truth.[16] | |
| Visualization of how to translate an English sentence into first-order logic | |
| Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid. In this example, the letter "c" represents Carmen while the letters "M" and "T" stand for "Mexican" and "teacher". The symbol "∧" has the meaning of "and". | |
| Formal logic uses formal languages to express, analyze, and clarify arguments.[17] They normally have a very limited vocabulary and exact syntactic rules. These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas.[18] This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid.[19] Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed.[20] | |
| The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning the rules of inference they accept as valid and the formal languages used to express them.[21] Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic.[22] For example, it has been suggested that only logically complete systems, like first-order logic, qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in the strict sense.[23] | |
| Informal logic | |
| Main article: Informal logic | |
| When understood in a wide sense, logic encompasses both formal and informal logic.[24] Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse.[25] Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments.[26] In this regard, it considers problems that formal logic on its own is unable to address.[27] Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies.[28] | |
| Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition.[29] The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language.[30] Formal logic can only examine them indirectly by translating them first into a formal language while informal logic investigates them in their original form.[31] On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) | |
| ∀ | |
| x | |
| ( | |
| B | |
| i | |
| r | |
| d | |
| ( | |
| x | |
| ) | |
| → | |
| F | |
| l | |
| i | |
| e | |
| s | |
| ( | |
| x | |
| ) | |
| ) | |
| {\displaystyle \forall x(Bird(x)\to Flies(x))}; (2) | |
| B | |
| i | |
| r | |
| d | |
| ( | |
| T | |
| w | |
| e | |
| e | |
| t | |
| y | |
| ) | |
| {\displaystyle Bird(Tweety)}; (3) | |
| F | |
| l | |
| i | |
| e | |
| s | |
| ( | |
| T | |
| w | |
| e | |
| e | |
| t | |
| y | |
| ) | |
| {\displaystyle Flies(Tweety)}" is studied by formal logic.[32] The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent.[33] Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of rationality, critical thinking, and the psychology of argumentation.[34] | |
| Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic.[35] Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to the conclusion "all ravens are black".[36] | |
| A further approach is to define informal logic as the study of informal fallacies.[37] Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument.[38] A false dilemma, for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us".[39] Some theorists state that formal logic studies the general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for the assessment of arguments.[40] | |
| Basic concepts | |
| Premises, conclusions, and truth | |
| Premises and conclusions | |
| Main articles: Premise and Logical consequence | |
| Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion.[41] For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers.[41][a] This means that they have a truth value: they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences.[43] Propositions are the denotations of sentences and are usually seen as abstract objects.[44] For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition.[45] | |
| Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions.[43] These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like the symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted.[46] Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism. It was discussed at length around the turn of the 20th century but it is not widely accepted today.[47] | |
| Internal structure | |
| Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex.[48] A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on the other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates.[49][48] For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by the propositional connective "and".[49] | |
| Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts.[49][50] But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects.[51] Whether the simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference.[52] | |
| Logical truth | |
| Main article: Logical truth | |
| Some complex propositions are true independently of the substantive meanings of their parts.[53] In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like the simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it.[54] This means that it is true under all interpretations of its non-logical terms. In some modal logics, this means that the proposition is true in all possible worlds.[55] Some theorists define logic as the study of logical truths.[16] | |
| Truth tables | |
| Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for the truth values "true" and "false".[56] The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " | |
| p | |
| ∧ | |
| q | |
| {\displaystyle p\land q}" uses the logical connective | |
| ∧ | |
| {\displaystyle \land } (and). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, | |
| p | |
| {\displaystyle p} ("yesterday was Sunday") and | |
| q | |
| {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are | |
| ¬ | |
| {\displaystyle \lnot } (not), | |
| ∨ | |
| {\displaystyle \lor } (or), | |
| → | |
| {\displaystyle \to } (if...then), and | |
| ↑ | |
| {\displaystyle \uparrow } (Sheffer stroke).[57] Given the conditional proposition | |
| p | |
| → | |
| q | |
| {\displaystyle p\to q}, one can form truth tables of its converse | |
| q | |
| → | |
| p | |
| {\displaystyle q\to p}, its inverse ( | |
| ¬ | |
| p | |
| → | |
| ¬ | |
| q | |
| {\displaystyle \lnot p\to \lnot q}), and its contrapositive ( | |
| ¬ | |
| q | |
| → | |
| ¬ | |
| p | |
| {\displaystyle \lnot q\to \lnot p}). Truth tables can also be defined for more complex expressions that use several propositional connectives.[58] | |
| Truth table of various expressions | |
| p q p ∧ q p ∨ q p → q ¬p → ¬q p | |
| ↑ | |
| {\displaystyle \uparrow } q | |
| T T T T T T F | |
| T F F T F T T | |
| F T F T T F T | |
| F F F F T T T | |
| Arguments and inferences | |
| Main articles: Argument and inference | |
| Logic is commonly defined in terms of arguments or inferences as the study of their correctness.[59] An argument is a set of premises together with a conclusion.[60] An inference is the process of reasoning from these premises to the conclusion.[43] But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on the other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises.[61] Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For a complex argument to be successful, each link of the chain has to be successful.[43] | |
| Diagram of argument terminology used in logic | |
| Argument terminology used in logic | |
| Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning.[62] The strongest form of support corresponds to deductive reasoning. But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases, the term ampliative or inductive reasoning is used.[63] Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic.[64] | |
| Deductive | |
| A deductively valid argument is one whose premises guarantee the truth of its conclusion.[11] For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs" is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises.[65] | |
| According to an influential view by Alfred Tarski, deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances.[66] | |
| Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference.[67] Rules of inference specify the form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid.[68] The modus ponens is a prominent rule of inference. It has the form "p; if p, then q; therefore q".[69] Knowing that it has just rained ( | |
| p | |
| {\displaystyle p}) and that after rain the streets are wet ( | |
| p | |
| → | |
| q | |
| {\displaystyle p\to q}), one can use modus ponens to deduce that the streets are wet ( | |
| q | |
| {\displaystyle q}).[70] | |
| The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false.[71] Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in the premises.[72] But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on the depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs.[73] | |
| Ampliative | |
| Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with a certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth.[74] This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility: it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn.[75] Ampliative reasoning plays a central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between.[76] | |
| The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term "induction" to cover all forms of non-deductive arguments.[77] But in a more narrow sense, induction is only one type of ampliative argument alongside abductive arguments.[78] Some philosophers, like Leo Groarke, also allow conductive arguments[b] as another type.[79] In this narrow sense, induction is often defined as a form of statistical generalization.[80] In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains.[81] In this sense, one may infer that "all elephants are gray" based on one's past observations of the color of elephants.[78] A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray.[81] Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference.[78] | |
| Abductive inference may or may not take statistical observations into consideration. In either case, the premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true.[82] In this sense, abduction is also called the inference to the best explanation.[83] For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean the table. This conclusion is justified because it is the best explanation of the current state of the kitchen.[78] For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it is not the best or most likely explanation.[82][83] | |
| Fallacies | |
| Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies. Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion.[84] So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though the conclusion is true. Some theorists, like John Stuart Mill, give a more restrictive definition of fallacies by additionally requiring that they appear to be correct.[85] This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them.[86] However, this reference to appearances is controversial because it belongs to the field of psychology, not logic, and because appearances may be different for different people.[87] | |
| Poster from 1901 | |
| Young America's dilemma: Shall I be wise and great, or rich and powerful? (poster from 1901). This is an example of a false dilemma: an informal fallacy using a disjunctive premise that excludes viable alternatives. | |
| Fallacies are usually divided into formal and informal fallacies.[38] For formal fallacies, the source of the error is found in the form of the argument. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore Othello is not male".[88] But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument.[89] Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity, the ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark".[90] Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise.[91] In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it.[92] | |
| Definitory and strategic rules | |
| The main focus of most logicians is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference.[93] They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach a given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess, for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king.[94] It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning.[93] | |
| Formal systems | |
| Main article: Formal system | |
| A formal system of logic consists of a formal language together with a set of axioms and a proof system used to draw inferences from these axioms.[95] In logic, axioms are statements that are accepted without proof. They are used to justify other statements.[96] Some theorists also include a semantics that specifies how the expressions of the formal language relate to real objects.[97] Starting in the late 19th century, many new formal systems have been proposed.[98] | |
| A formal language consists of an alphabet and syntactic rules. The alphabet is the set of basic symbols used in expressions. The syntactic rules determine how these symbols may be arranged to result in well-formed formulas.[99] For instance, the syntactic rules of propositional logic determine that " | |
| P | |
| ∧ | |
| Q | |
| {\displaystyle P\land Q}" is a well-formed formula but " | |
| ∧ | |
| Q | |
| {\displaystyle \land Q}" is not since the logical conjunction | |
| ∧ | |
| {\displaystyle \land } requires terms on both sides.[100] | |
| A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of conjunction introduction states that | |
| P | |
| ∧ | |
| Q | |
| {\displaystyle P\land Q} follows from the premises | |
| P | |
| {\displaystyle P} and | |
| Q | |
| {\displaystyle Q}. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including natural deduction and sequent calculi.[101] | |
| A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance, the semantics for classical propositional logic assigns the formula | |
| P | |
| ∧ | |
| Q | |
| {\displaystyle P\land Q} the denotation "true" whenever | |
| P | |
| {\displaystyle P} and | |
| Q | |
| {\displaystyle Q} are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true.[102] | |
| A system of logic is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly.[103] | |
| Systems of logic | |
| Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western world,[104] but modern developments in this field have led to a vast proliferation of logical systems.[105] One prominent categorization divides modern formal logical systems into classical logic, extended logics, and deviant logics.[106] | |
| Aristotelian | |
| Main article: Aristotelian logic | |
| Aristotelian logic encompasses a great variety of topics. They include metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject, a predicate, and a copula connecting the subject to the predicate.[107] For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is".[108] The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using a logical connective like "and" to form a new complex proposition.[109] | |
| Diagram of the square of opposition | |
| The square of opposition is often used to visualize the relations between the four basic categorical propositions in Aristotelian logic. It shows, for example, that the propositions "All S are P" and "Some S are not P" are contradictory, meaning that one of them has to be true while the other is false. | |
| In Aristotelian logic, the subject can be universal, particular, indefinite, or singular. For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates".[110] | |
| Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities.[111] The predicate can be linked to the subject in two ways: either by affirming it or by denying it.[112] For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one term in each case.[113] Thus, these three propositions contain three terms, referred to as major term, minor term, and middle term.[114] The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed.[112][115] For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid.[116] | |
| Classical | |
| Main article: Classical logic | |
| Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in the sense that it is based on basic logical intuitions shared by most logicians.[117] These intuitions include the law of excluded middle, the double negation elimination, the principle of explosion, and the bivalence of truth.[118] It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are the contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future.[119] Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond the scope of mathematics.[120] | |
| Propositional logic | |
| Main article: Propositional calculus | |
| Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives. For instance, propositional logic represents the conjunction of two atomic propositions | |
| P | |
| {\displaystyle P} and | |
| Q | |
| {\displaystyle Q} as the complex formula | |
| P | |
| ∧ | |
| Q | |
| {\displaystyle P\land Q}. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component.[121] Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition.[122] | |
| First-order logic | |
| Symbol introduced by Gottlob Frege for the universal quantifier | |
| Gottlob Frege's Begriffsschrift introduced the notion of quantifier in a graphical notation, which here represents the judgment that | |
| ∀ | |
| x | |
| . | |
| F | |
| ( | |
| x | |
| ) | |
| {\displaystyle \forall x.F(x)} is true. | |
| Main article: First-order logic | |
| First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates, which refer to properties and relations, and quantifiers, which treat notions like "some" and "all".[123] For example, to express the proposition "this raven is black", one may use the predicate | |
| B | |
| {\displaystyle B} for the property "black" and the singular term | |
| r | |
| {\displaystyle r} referring to the raven to form the expression | |
| B | |
| ( | |
| r | |
| ) | |
| {\displaystyle B(r)}. To express that some objects are black, the existential quantifier | |
| ∃ | |
| {\displaystyle \exists } is combined with the variable | |
| x | |
| {\displaystyle x} to form the proposition | |
| ∃ | |
| x | |
| B | |
| ( | |
| x | |
| ) | |
| {\displaystyle \exists xB(x)}. First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer | |
| ∃ | |
| x | |
| B | |
| ( | |
| x | |
| ) | |
| {\displaystyle \exists xB(x)} from | |
| B | |
| ( | |
| r | |
| ) | |
| {\displaystyle B(r)}.[124] | |
| Extended | |
| Extended logics are logical systems that accept the basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics, ethics, and epistemology.[125] | |
| Modal logic | |
| Main article: Modal logic | |
| Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: | |
| ◊ | |
| {\displaystyle \Diamond } expresses that something is possible while | |
| ◻ | |
| {\displaystyle \Box } expresses that something is necessary.[126] For example, if the formula | |
| B | |
| ( | |
| s | |
| ) | |
| {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula | |
| ◊ | |
| B | |
| ( | |
| s | |
| ) | |
| {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker".[127] To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that | |
| ◊ | |
| A | |
| {\displaystyle \Diamond A} follows from | |
| ◻ | |
| A | |
| {\displaystyle \Box A}. Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that | |
| ◻ | |
| A | |
| {\displaystyle \Box A} is equivalent to | |
| ¬ | |
| ◊ | |
| ¬ | |
| A | |
| {\displaystyle \lnot \Diamond \lnot A}.[128] | |
| Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns the field of ethics and introduces symbols to express the ideas of obligation and permission, i.e. to describe whether an agent has to perform a certain action or is allowed to perform it.[129] The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time.[129] In epistemology, epistemic modal logic is used to represent the ideas of knowing something in contrast to merely believing it to be the case.[130] | |
| Higher order logic | |
| Main article: Higher-order logic | |
| Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification.[131] Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula " | |
| ∃ | |
| x | |
| ( | |
| A | |
| p | |
| p | |
| l | |
| e | |
| ( | |
| x | |
| ) | |
| ∧ | |
| S | |
| w | |
| e | |
| e | |
| t | |
| ( | |
| x | |
| ) | |
| ) | |
| {\displaystyle \exists x(Apple(x)\land Sweet(x))}" (some apples are sweet) is an example of the existential quantifier " | |
| ∃ | |
| {\displaystyle \exists }" applied to the individual variable " | |
| x | |
| {\displaystyle x}". In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula " | |
| ∃ | |
| Q | |
| ( | |
| Q | |
| ( | |
| M | |
| a | |
| r | |
| y | |
| ) | |
| ∧ | |
| Q | |
| ( | |
| J | |
| o | |
| h | |
| n | |
| ) | |
| ) | |
| {\displaystyle \exists Q(Q(Mary)\land Q(John))}". In this case, the existential quantifier is applied to the predicate variable " | |
| Q | |
| {\displaystyle Q}".[132] The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories.[43] But it has drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still more commonly used.[133] | |
| Deviant | |
| Main article: Deviant logic | |
| Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue.[134] | |
| Intuitionistic logic is a restricted version of classical logic.[135] It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that | |
| A | |
| {\displaystyle A} follows from | |
| ¬ | |
| ¬ | |
| A | |
| {\displaystyle \lnot \lnot A}. This is a valid rule of inference in classical logic but it is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle. It states that for every sentence, either it or its negation is true. This means that every proposition of the form | |
| A | |
| ∨ | |
| ¬ | |
| A | |
| {\displaystyle A\lor \lnot A} is true.[135] These deviations from classical logic are based on the idea that truth is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics, which emphasizes the need to find or construct a specific example to prove its existence.[136] | |
| Multi-valued logics depart from classicality by rejecting the principle of bivalence, which requires all propositions to be either true or false. For instance, Jan Łukasiewicz and Stephen Cole Kleene both proposed ternary logics which have a third truth value representing that a statement's truth value is indeterminate.[137] These logics have been applied in the field of linguistics. Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a real number between 0 and 1.[138] | |
| Paraconsistent logics are logical systems that can deal with contradictions. They are formulated to avoid the principle of explosion: for them, it is not the case that anything follows from a contradiction.[139] They are often motivated by dialetheism, the view that contradictions are real or that reality itself is contradictory. Graham Priest is an influential contemporary proponent of this position and similar views have been ascribed to Georg Wilhelm Friedrich Hegel.[140] | |
| Informal | |
| Main article: Informal logic | |
| Informal logic is usually carried out in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation. Nonetheless, some frameworks of informal logic have also been presented that try to provide a systematic characterization of the correctness of arguments.[141] | |
| The pragmatic or dialogical approach to informal logic sees arguments as speech acts and not merely as a set of premises together with a conclusion.[142] As speech acts, they occur in a certain context, like a dialogue, which affects the standards of right and wrong arguments.[143] A prominent version by Douglas N. Walton understands a dialogue as a game between two players. The initial position of each player is characterized by the propositions to which they are committed and the conclusion they intend to prove. Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion.[144] This is achieved by making arguments: arguments are the moves of the game.[145] They affect to which propositions the players are committed. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them. This is usually not possible straight away. For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one's intended conclusion. Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent's victory by denying their conclusion.[144] Whether an argument is correct depends on whether it promotes the progress of the dialogue. Fallacies, on the other hand, are violations of the standards of proper argumentative rules.[146] These standards also depend on the type of dialogue. For example, the standards governing the scientific discourse differ from the standards in business negotiations.[147] | |
| The epistemic approach to informal logic, on the other hand, focuses on the epistemic role of arguments.[148] It is based on the idea that arguments aim to increase our knowledge. They achieve this by linking justified beliefs to beliefs that are not yet justified.[149] Correct arguments succeed at expanding knowledge while fallacies are epistemic failures: they do not justify the belief in their conclusion.[150] For example, the fallacy of begging the question is a fallacy because it fails to provide independent justification for its conclusion, even though it is deductively valid.[151] In this sense, logical normativity consists in epistemic success or rationality.[149] The Bayesian approach is one example of an epistemic approach.[152] Central to Bayesianism is not just whether the agent believes something but the degree to which they believe it, the so-called credence. Degrees of belief are seen as subjective probabilities in the believed proposition, i.e. how certain the agent is that the proposition is true.[153] On this view, reasoning can be interpreted as a process of changing one's credences, often in reaction to new incoming information.[154] Correct reasoning and the arguments it is based on follow the laws of probability, for example, the principle of conditionalization. Bad or irrational reasoning, on the other hand, violates these laws.[155] | |
| Areas of research | |
| Logic is studied in various fields. In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science.[156] In other cases, logic itself is made the subject of research in another discipline. This can happen in diverse ways. For instance, it can involve investigating the philosophical assumptions linked to the basic concepts used by logicians. Other ways include interpreting and analyzing logic through mathematical structures as well as studying and comparing abstract properties of formal logical systems.[157] | |
| Philosophy of logic and philosophical logic | |
| Main articles: Philosophy of logic and Philosophical logic | |
| Philosophy of logic is the philosophical discipline studying the scope and nature of logic.[59] It examines many presuppositions implicit in logic, like how to define its basic concepts or the metaphysical assumptions associated with them.[158] It is also concerned with how to classify logical systems and considers the ontological commitments they incur.[159] Philosophical logic is one of the areas within the philosophy of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics, ethics, and epistemology.[160] This application usually happens in the form of extended or deviant logical systems.[161] | |
| Metalogic | |
| Main article: Metalogic | |
| Metalogic is the field of inquiry studying the properties of formal logical systems. For example, when a new formal system is developed, metalogicians may study it to determine which formulas can be proven in it. They may also study whether an algorithm could be developed to find a proof for each formula and whether every provable formula in it is a tautology. Finally, they may compare it to other logical systems to understand its distinctive features. A key issue in metalogic concerns the relation between syntax and semantics. The syntactic rules of a formal system determine how to deduce conclusions from premises, i.e. how to formulate proofs. The semantics of a formal system governs which sentences are true and which ones are false. This determines the validity of arguments since, for valid arguments, it is impossible for the premises to be true and the conclusion to be false. The relation between syntax and semantics concerns issues like whether every valid argument is provable and whether every provable argument is valid. Metalogicians also study whether logical systems are complete, sound, and consistent. They are interested in whether the systems are decidable and what expressive power they have. Metalogicians usually rely heavily on abstract mathematical reasoning when examining and formulating metalogical proofs. This way, they aim to arrive at precise and general conclusions on these topics.[162] | |
| Mathematical logic | |
| Main article: Mathematical logic | |
| Photograph of Bertrand Russell | |
| Bertrand Russell made various contributions to mathematical logic.[163] | |
| The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory.[164] Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics.[165] The latter was a major concern in early 20th-century mathematical logic, which pursued the program of logicism pioneered by philosopher-logicians such as Gottlob Frege, Alfred North Whitehead, and Bertrand Russell. Mathematical theories were supposed to be logical tautologies, and their program was to show this by means of a reduction of mathematics to logic. Many attempts to realize this program failed, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems.[166] | |
| Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic. They include Cantor's theorem, the status of the Axiom of Choice, the question of the independence of the continuum hypothesis, and the modern debate on large cardinal axioms.[167] | |
| Computability theory is the branch of mathematical logic that studies effective procedures to solve calculation problems. One of its main goals is to understand whether it is possible to solve a given problem using an algorithm. For instance, given a certain claim about the positive integers, it examines whether an algorithm can be found to determine if this claim is true. Computability theory uses various theoretical tools and models, such as Turing machines, to explore this type of issue.[168] | |
| Computational logic | |
| Main articles: Computational logic and Logic in computer science | |
| Diagram of an AND gate using transistors | |
| Conjunction (AND) is one of the basic operations of Boolean logic. It can be electronically implemented in several ways, for example, by using two transistors. | |
| Computational logic is the branch of logic and computer science that studies how to implement mathematical reasoning and logical formalisms using computers. This includes, for example, automatic theorem provers, which employ rules of inference to construct a proof step by step from a set of premises to the intended conclusion without human intervention.[169] Logic programming languages are designed specifically to express facts using logical formulas and to draw inferences from these facts. For example, Prolog is a logic programming language based on predicate logic.[170] Computer scientists also apply concepts from logic to problems in computing. The works of Claude Shannon were influential in this regard. He showed how Boolean logic can be used to understand and implement computer circuits.[171] This can be achieved using electronic logic gates, i.e. electronic circuits with one or more inputs and usually one output. The truth values of propositions are represented by voltage levels. In this way, logic functions can be simulated by applying the corresponding voltages to the inputs of the circuit and determining the value of the function by measuring the voltage of the output.[172] | |
| Formal semantics of natural language | |
| Main article: Formal semantics (natural language) | |
| Formal semantics is a subfield of logic, linguistics, and the philosophy of language. The discipline of semantics studies the meaning of language. Formal semantics uses formal tools from the fields of symbolic logic and mathematics to give precise theories of the meaning of natural language expressions. It understands meaning usually in relation to truth conditions, i.e. it examines in which situations a sentence would be true or false. One of its central methodological assumptions is the principle of compositionality. It states that the meaning of a complex expression is determined by the meanings of its parts and how they are combined. For example, the meaning of the verb phrase "walk and sing" depends on the meanings of the individual expressions "walk" and "sing". Many theories in formal semantics rely on model theory. This means that they employ set theory to construct a model and then interpret the meanings of expression in relation to the elements in this model. For example, the term "walk" may be interpreted as the set of all individuals in the model that share the property of walking. Early influential theorists in this field were Richard Montague and Barbara Partee, who focused their analysis on the English language.[173] | |
| Epistemology of logic | |
| The epistemology of logic studies how one knows that an argument is valid or that a proposition is logically true.[174] This includes questions like how to justify that modus ponens is a valid rule of inference or that contradictions are false.[175] The traditionally dominant view is that this form of logical understanding belongs to knowledge a priori.[176] In this regard, it is often argued that the mind has a special faculty to examine relations between pure ideas and that this faculty is also responsible for apprehending logical truths.[177] A similar approach understands the rules of logic in terms of linguistic conventions. On this view, the laws of logic are trivial since they are true by definition: they just express the meanings of the logical vocabulary.[178] | |
| Some theorists, like Hilary Putnam and Penelope Maddy, object to the view that logic is knowable a priori. They hold instead that logical truths depend on the empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world. According to this view, they may be explored by studying general patterns of the fundamental sciences. For example, it has been argued that certain insights of quantum mechanics refute the principle of distributivity in classical logic, which states that the formula | |
| A | |
| ∧ | |
| ( | |
| B | |
| ∨ | |
| C | |
| ) | |
| {\displaystyle A\land (B\lor C)} is equivalent to | |
| ( | |
| A | |
| ∧ | |
| B | |
| ) | |
| ∨ | |
| ( | |
| A | |
| ∧ | |
| C | |
| ) | |
| {\displaystyle (A\land B)\lor (A\land C)}. This claim can be used as an empirical argument for the thesis that quantum logic is the correct logical system and should replace classical logic.[179] | |
| History | |
| Main article: History of logic | |
| Bust of Aristotle | |
| Portrait of Avicenna | |
| Portrait of William of Ockham | |
| Bust showing Gottlob Frege | |
| Top row: Aristotle, who established the canon of western philosophy;[108] and Avicenna, who replaced Aristotelian logic in Islamic discourse.[180] Bottom row: William of Ockham, a major figure of medieval scholarly thought;[181] and Gottlob Frege, one of the founders of modern symbolic logic.[182] | |
| Logic was developed independently in several cultures during antiquity. One major early contributor was Aristotle, who developed term logic in his Organon and Prior Analytics.[183] He was responsible for the introduction of the hypothetical syllogism[184] and temporal modal logic.[185] Further innovations include inductive logic[186] as well as the discussion of new logical concepts such as terms, predicables, syllogisms, and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century.[187] It has now been superseded by later work, though many of its key insights are still present in modern systems of logic.[188] | |
| Ibn Sina (Avicenna) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world.[189] It influenced Western medieval writers such as Albertus Magnus and William of Ockham.[190] Ibn Sina wrote on the hypothetical syllogism[191] and on the propositional calculus.[192] He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic.[193] He also made use of inductive logic, such as his methods of agreement, difference, and concomitant variation, which are critical to the scientific method.[191] Fakhr al-Din al-Razi was another influential Muslim logician. He criticized Aristotelian syllogistics and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill.[194] | |
| During the Middle Ages, many translations and interpretations of Aristotelian logic were made. The works of Boethius were particularly influential. Besides translating Aristotle's work into Latin, he also produced textbooks on logic.[195] Later, the works of Islamic philosophers such as Ibn Sina and Ibn Rushd (Averroes) were drawn on. This expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323, William of Ockham's influential Summa Logicae was released. It is a comprehensive treatise on logic that discusses many basic concepts of logic and provides a systematic exposition of types of propositions and their truth conditions.[196] | |
| In Chinese philosophy, the School of Names and Mohism were particularly influential. The School of Names focused on the use of language and on paradoxes. For example, Gongsun Long proposed the white horse paradox, which defends the thesis that a white horse is not a horse. The school of Mohism also acknowledged the importance of language for logic and tried to relate the ideas in these fields to the realm of ethics.[197] | |
| In India, the study of logic was primarily pursued by the schools of Nyaya, Buddhism, and Jainism. It was not treated as a separate academic discipline and discussions of its topics usually happened in the context of epistemology and theories of dialogue or argumentation.[198] In Nyaya, inference is understood as a source of knowledge (pramāṇa). It follows the perception of an object and tries to arrive at conclusions, for example, about the cause of this object.[199] A similar emphasis on the relation to epistemology is also found in Buddhist and Jainist schools of logic, where inference is used to expand the knowledge gained through other sources.[200] Some of the later theories of Nyaya, belonging to the Navya-Nyāya school, resemble modern forms of logic, such as Gottlob Frege's distinction between sense and reference and his definition of number.[201] | |
| The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of modern symbolic logic.[202] Many see Gottlob Frege's Begriffsschrift as the birthplace of modern logic. Gottfried Wilhelm Leibniz's idea of a universal formal language is often considered a forerunner. Other pioneers were George Boole, who invented Boolean algebra as a mathematical system of logic, and Charles Peirce, who developed the logic of relatives. Alfred North Whitehead and Bertrand Russell, in turn, condensed many of these insights in their work Principia Mathematica. Modern logic introduced novel concepts, such as functions, quantifiers, and relational predicates. A hallmark of modern symbolic logic is its use of formal language to precisely codify its insights. In this regard, it departs from earlier logicians, who relied mainly on natural language.[203] Of particular influence was the development of first-order logic, which is usually treated as the standard system of modern logic.[204] Its analytical generality allowed the formalization of mathematics and drove the investigation of set theory. It also made Alfred Tarski's approach to model theory possible and provided the foundation of modern mathematical logic.[205] |