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137-444: Parameter has more specific meanings within various disciplines, including mathematics , computer programming , engineering , statistics , logic , linguistics , and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. When

274-410: A model describes the probability that something will occur. Parameters in a model are the weight of the various probabilities. Tiernan Ray, in an article on GPT-3, described parameters this way: A parameter is a calculation in a neural network that applies a great or lesser weighting to some aspect of the data, to give that aspect greater or lesser prominence in the overall calculation of the data. It

411-408: A parametric equation this can be written The parameter t in this equation would elsewhere in mathematics be called the independent variable . In mathematical analysis , integrals dependent on a parameter are often considered. These are of the form In this formula, t is the argument of the function F , and on the right-hand side the parameter on which the integral depends. When evaluating

548-591: 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 . 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: The study of types of algebraic structures as mathematical objects

685-453: A system is modeled by equations, the values that describe the system are called parameters . For example, in mechanics , the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities (for fluids), appear as parameters in the equations modeling movements. There are often several choices for the parameters, and choosing a convenient set of parameters is called parametrization . For example, if one were considering

822-406: A (relatively) small area, like within a particular country or region. Such parametrizations are also relevant to the modelization of geographic areas (i.e. map drawing ). Mathematical functions have one or more arguments that are designated in the definition by variables . A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that

959-428: A broader result in closed monoidal categories : Currying is the statement that the tensor product and the internal Hom are adjoint functors ; that is, for every object B {\displaystyle B} there is a natural isomorphism : Here, Hom denotes the (external) Hom-functor of all morphisms in the category, while B ⇒ C {\displaystyle B\Rightarrow C} denotes

1096-402: A category is not closed , and thus lacks an internal hom functor (possibly because there is more than one choice for such a functor). Another way is if it is not monoidal , and thus lacks a product (that is, lacks a way of writing down pairs of objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesian closed categories

1233-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

1370-669: 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)

1507-540: A function currying constructs a new function That is, g {\displaystyle g} takes an argument of type X {\displaystyle X} and returns a function of type Y → Z {\displaystyle Y\to Z} . It is defined by for x {\displaystyle x} of type X {\displaystyle X} and y {\displaystyle y} of type Y {\displaystyle Y} . We then also write Uncurrying

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1644-478: A function with more than two arguments can be defined by induction. Currying is useful in both practical and theoretical settings. In functional programming languages , and many others, it provides a way of automatically managing how arguments are passed to functions and exceptions. In theoretical computer science , it provides a way to study functions with multiple arguments in simpler theoretical models which provide only one argument. The most general setting for

1781-582: A function, that takes f {\displaystyle f} as an argument, and returns a function that maps each x {\displaystyle x} to f x . {\displaystyle f_{x}.} The proper notation for expressing this is verbose. The function f {\displaystyle f} belongs to the set of functions ( X × Y ) → Z . {\displaystyle (X\times Y)\to Z.} Meanwhile, f x {\displaystyle f_{x}} belongs to

1918-425: A homotopy of two functions from Y {\displaystyle Y} to Z {\displaystyle Z} , or, equivalently, a single (continuous) path in Z Y {\displaystyle Z^{Y}} . In algebraic topology , currying serves as an example of Eckmann–Hilton duality , and, as such, plays an important role in a variety of different settings. For example, loop space

2055-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

2192-422: 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

2329-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . 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

2466-409: A natural complement to the language of category theory , as discussed below. This is because categories, and specifically, monoidal categories , have an internal language , with simply-typed lambda calculus being the most prominent example of such a language. It is important in this context, because it can be built from a single type constructor, the arrow type. Currying then endows the language with

2603-890: A natural product type. The correspondence between objects in categories and types then allows programming languages to be re-interpreted as logics (via Curry–Howard correspondence ), and as other types of mathematical systems, as explored further, below. Under the Curry–Howard correspondence , the existence of currying and uncurrying is equivalent to the logical theorem ( ( A ∧ B ) → C ) ⇔ ( A → ( B → C ) ) {\displaystyle ((A\land B)\to C)\Leftrightarrow (A\to (B\to C))} (also known as exportation ), as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication. The exponential object Q P {\displaystyle Q^{P}} in

2740-425: A parameter denotes an element which may be manipulated (composed), separately from the other elements. The term is used particularly for pitch , loudness , duration , and timbre , though theorists or composers have sometimes considered other musical aspects as parameters. The term is particularly used in serial music , where each parameter may follow some specified series. Paul Lansky and George Perle criticized

2877-426: A polynomial function of k (when n is considered a parameter). Indeed, in the latter case, it is only defined for non-negative integer arguments. More formal presentations of such situations typically start out with a function of several variables (including all those that might sometimes be called "parameters") such as as the most fundamental object being considered, then defining functions with fewer variables from

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3014-535: A population mean with a given level of confidence. 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 . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

3151-399: A probability distribution: see Statistical parameter . In computer programming , two notions of parameter are commonly used, and are referred to as parameters and arguments —or more formally as a formal parameter and an actual parameter . For example, in the definition of a function such as x is the formal parameter (the parameter ) of the defined function. When the function

3288-453: A property characteristic of the system. k is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample. If we want to know the probability of observing k 1 occurrences, we plug it into the function to get f ( k 1 ; λ ) {\displaystyle f(k_{1};\lambda )} . Without altering the system, we can take multiple samples, which will have

3425-445: A proportion given by the probability mass function above. From measurement to measurement, however, λ remains constant at 5. If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase. Another common distribution is the normal distribution , which has as parameters

3562-427: A range of values of k , but the system is always characterized by the same λ. For instance, suppose we have a radioactive sample that emits, on average, five particles every ten minutes. We take measurements of how many particles the sample emits over ten-minute periods. The measurements exhibit different values of k , and if the sample behaves according to Poisson statistics, then each value of k will come up in

3699-428: A result, the application of f {\displaystyle f} and subsequently, g {\displaystyle g} , to those arguments. The process can be iterated. Currying provides a way for working with functions that take multiple arguments, and using them in frameworks where functions might take only one argument. For example, some analytical techniques can only be applied to functions with

3836-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

3973-424: 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. The concept of algebraic structure addresses this, consisting of

4110-528: A single argument. Practical functions frequently take more arguments than this. Frege showed that it was sufficient to provide solutions for the single argument case, as it was possible to transform a function with multiple arguments into a chain of single-argument functions instead. This transformation is the process now known as currying. All "ordinary" functions that might typically be encountered in mathematical analysis or in computer programming can be curried. However, there are categories in which currying

4247-418: 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

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4384-477: 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 . 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

4521-703: 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 . 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. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

4658-708: Is flat " and "a field is always a ring ". Currying In mathematics and computer science , currying is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In the prototypical example, one begins with a function f : ( X × Y ) → Z {\displaystyle f:(X\times Y)\to Z} that takes two arguments, one from X {\displaystyle X} and one from Y , {\displaystyle Y,} and produces objects in Z . {\displaystyle Z.} The curried form of this function treats

4795-558: Is a type constructor , specifically, the function type or arrow type. Similarly, the Cartesian product X × Y {\displaystyle X\times Y} of types is constructed by the product type constructor × {\displaystyle \times } . The type-theoretical approach is expressed in programming languages such as ML and the languages derived from and inspired by it: Caml , Haskell , and F# . The type-theoretical approach provides

4932-524: Is a universal property of an exponential object , and gives rise to an adjunction in cartesian closed categories . That is, there is a natural isomorphism between the morphisms from a binary product f : ( X × Y ) → Z {\displaystyle f\colon (X\times Y)\to Z} and the morphisms to an exponential object g : X → Z Y {\displaystyle g\colon X\to Z^{Y}} . This generalizes to

5069-471: Is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as

5206-434: Is a function that takes objects from the first set, and returns objects in the second set, and so one writes curry : [ ( X × Y ) → Z ] → ( X → [ Y → Z ] ) . {\displaystyle {\mbox{curry}}:[(X\times Y)\to Z]\to (X\to [Y\to Z]).} This is a somewhat informal example; more precise definitions of what

5343-402: Is a matter of convention (or historical accident) whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object. For instance, the notation for the falling factorial power defines a polynomial function of n (when k is considered a parameter), but is not

5480-421: Is adjoint to reduced suspensions ; this is commonly written as where [ A , B ] {\displaystyle [A,B]} is the set of homotopy classes of maps A → B {\displaystyle A\rightarrow B} , and Σ A {\displaystyle \Sigma A} is the suspension of A , and Ω A {\displaystyle \Omega A}

5617-641: Is being considered. A parameter could be incorporated into the function name to indicate its dependence on the parameter. For instance, one may define the base- b logarithm by the formula where b is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, be a constant when considering the derivative log b ′ ⁡ ( x ) = ( x ln ⁡ ( b ) ) − 1 {\displaystyle \textstyle \log _{b}'(x)=(x\ln(b))^{-1}} . In some informal situations it

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5754-477: Is commonly interested in continuous functions between topological spaces . One writes Hom ( X , Y ) {\displaystyle {\text{Hom}}(X,Y)} (the Hom functor ) for the set of all functions from X {\displaystyle X} to Y {\displaystyle Y} , and uses the notation Y X {\displaystyle Y^{X}} to denote

5891-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

6028-554: Is continuous if and only if its curried form is continuous. Another important result is that the application map , usually called "evaluation" in this context, is continuous (note that eval is a strictly different concept in computer science.) That is, eval : Y X × X → Y ( f , x ) ↦ f ( x ) {\displaystyle {\begin{aligned}&&{\text{eval}}:Y^{X}\times X\to Y\\&&(f,x)\mapsto f(x)\end{aligned}}}

6165-572: Is continuous when Y X {\displaystyle Y^{X}} is compact-open and Y {\displaystyle Y} locally compact Hausdorff. These two results are central for establishing the continuity of homotopy , i.e. when X {\displaystyle X} is the unit interval I {\displaystyle I} , so that Z I × Y ≅ ( Z Y ) I {\displaystyle Z^{I\times Y}\cong (Z^{Y})^{I}} can be thought of as either

6302-513: Is defined by the set of all similar objects and the properties that these objects must have. 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. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

6439-407: 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. 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

6576-399: Is equivalent to That is, the parenthesis are not required to disambiguate the order of the application. Curried functions may be used in any programming language that supports closures ; however, uncurried functions are generally preferred for efficiency reasons, since the overhead of partial application and closure creation can then be avoided for most function calls. In type theory ,

6713-712: Is evaluated for a given value, as in 3 is the actual parameter (the argument ) for evaluation by the defined function; it is a given value (actual value) that is substituted for the formal parameter of the defined function. (In casual usage the terms parameter and argument might inadvertently be interchanged, and thereby used incorrectly.) These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic . Terminology varies between languages; some computer languages such as C define parameter and argument as given here, while Eiffel uses an alternative convention . In artificial intelligence ,

6850-493: 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. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

6987-451: Is meant by "object" and "function" are given below. These definitions vary from context to context, and take different forms, depending on the theory that one is working in. Currying is related to, but not the same as, partial application . The example above can be used to illustrate partial application; it is quite similar. Partial application is the function apply {\displaystyle {\mbox{apply}}} that takes

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7124-591: Is mentioned later in the context of higher-order functions. John C. Reynolds defined "currying" in a 1972 paper, but did not claim to have coined the term. Currying is most easily understood by starting with an informal definition, which can then be molded to fit many different domains. First, there is some notation to be established. The notation X → Y {\displaystyle X\to Y} denotes all functions from X {\displaystyle X} to Y {\displaystyle Y} . If f {\displaystyle f}

7261-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. 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 . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

7398-437: Is not possible; the most general categories which allow currying are the closed monoidal categories . Some programming languages almost always use curried functions to achieve multiple arguments; notable examples are ML and Haskell , where in both cases all functions have exactly one argument. This property is inherited from lambda calculus , where multi-argument functions are usually represented in curried form. Currying

7535-404: 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,

7672-1192: 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." 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. 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 , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

7809-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . 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. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

7946-433: 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. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

8083-461: Is possible to make statistical inferences without assuming a particular parametric family of probability distributions . In that case, one speaks of non-parametric statistics as opposed to the parametric statistics just described. For example, a test based on Spearman's rank correlation coefficient would be called non-parametric since the statistic is computed from the rank-order of the data disregarding their actual values (and thus regardless of

8220-490: Is related to, but not the same as partial application . In practice, the programming technique of closures can be used to perform partial application and a kind of currying, by hiding arguments in an environment that travels with the curried function. The "Curry" in "Currying" is a reference to logician Haskell Curry , who used the concept extensively, but Moses Schönfinkel had the idea six years before Curry. The alternative name "Schönfinkelisation" has been proposed. In

8357-567: Is sometimes mistranslated as a condemnation of mathematicians. 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

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8494-748: Is such a function, we write f : X → Y {\displaystyle f\colon X\to Y} . Let X × Y {\displaystyle X\times Y} denote the ordered pairs of the elements of X {\displaystyle X} and Y {\displaystyle Y} respectively, that is, the Cartesian product of X {\displaystyle X} and Y {\displaystyle Y} . Here, X {\displaystyle X} and Y {\displaystyle Y} may be sets, or they may be types, or they may be other kinds of objects, as explored below. Given

8631-437: Is sufficient for the discussion of classical logic ; the more general setting of closed monoidal categories is suitable for quantum computation . The difference between these two is that the product for cartesian categories (such as the category of sets , complete partial orders or Heyting algebras ) is just the Cartesian product ; it is interpreted as an ordered pair of items (or a list). Simply typed lambda calculus

8768-464: Is the adjoint functor that maps suspensions to loop spaces, and uncurrying is the dual. The duality between the mapping cone and the mapping fiber ( cofibration and fibration ) can be understood as a form of currying, which in turn leads to the duality of the long exact and coexact Puppe sequences . In homological algebra , the relationship between currying and uncurrying is known as tensor-hom adjunction . Here, an interesting twist arises:

8905-497: Is the dual transformation to currying, and can be seen as a form of defunctionalization . It takes a function f {\displaystyle f} whose return value is another function g {\displaystyle g} , and yields a new function f ′ {\displaystyle f'} that takes as parameters the arguments for both f {\displaystyle f} and g {\displaystyle g} , and returns, as

9042-664: Is the loop space of A . In essence, the suspension Σ X {\displaystyle \Sigma X} can be seen as the cartesian product of X {\displaystyle X} with the unit interval, modulo an equivalence relation to turn the interval into a loop. The curried form then maps the space X {\displaystyle X} to the space of functions from loops into Z {\displaystyle Z} , that is, from X {\displaystyle X} into Ω Z {\displaystyle \Omega Z} . Then curry {\displaystyle {\text{curry}}}

9179-442: Is the natural bijection between the set A B × C {\displaystyle A^{B\times C}} of functions from B × C {\displaystyle B\times C} to A {\displaystyle A} , and the set ( A C ) B {\displaystyle (A^{C})^{B}} of functions from B {\displaystyle B} to

9316-418: Is the purpose of universal algebra and category theory . 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 . Calculus, formerly called infinitesimal calculus,

9453-445: Is the reverse transformation, and is most easily understood in terms of its right adjoint, the function apply . {\displaystyle \operatorname {apply} .} In set theory , the notation Y X {\displaystyle Y^{X}} is used to denote the set of functions from the set X {\displaystyle X} to the set Y {\displaystyle Y} . Currying

9590-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

9727-507: Is these weights that give shape to the data, and give the neural network a learned perspective on the data. In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. This usage is not consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel. "Speaking generally, properties are those physical quantities which directly describe

9864-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

10001-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. 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

10138-768: 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. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

10275-592: The Hom functor and the tensor product functor might not lift to an exact sequence ; this leads to the definition of the Ext functor and the Tor functor . In order theory , that is, the theory of lattices of partially ordered sets , curry {\displaystyle {\text{curry}}} is a continuous function when the lattice is given the Scott topology . Scott-continuous functions were first investigated in

10412-511: The Pythagoreans appeared to have considered it a subject in its own right. 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. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

10549-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

10686-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

10823-440: The lambda calculus , in which functions only take a single argument. Consider a function f ( x , y ) {\displaystyle f(x,y)} taking two arguments, and having the type ( X × Y ) → Z {\displaystyle (X\times Y)\to Z} , which should be understood to mean that x must have the type X {\displaystyle X} , y must have

10960-400: 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

11097-405: 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

11234-532: The 20th century. 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. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

11371-637: The Middle Ages and made available in Europe. 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 ,

11508-513: The Scott topology is typically finer , and is not sober . The notion of continuity makes its appearance in homotopy type theory , where, roughly speaking, two computer programs can be considered to be homotopic, i.e. compute the same results, if they can be "continuously" refactored from one to the other. In theoretical computer science , currying provides a way to study functions with multiple arguments in very simple theoretical models, such as

11645-403: The attempt to provide a semantics for lambda calculus (as ordinary set theory is inadequate to do this). More generally, Scott-continuous functions are now studied in domain theory , which encompasses the study of denotational semantics of computer algorithms. Note that the Scott topology is quite different than many common topologies one might encounter in the category of topological spaces ;

11782-583: The beginnings of algebra (Diophantus, 3rd 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 . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

11919-548: The category of Heyting algebras is normally written as material implication P → Q {\displaystyle P\to Q} . Distributive Heyting algebras are Boolean algebras , and the exponential object has the explicit form ¬ P ∨ Q {\displaystyle \neg P\lor Q} , thus making it clear that the exponential object really is material implication . The above notions of currying and uncurrying find their most general, abstract statement in category theory . Currying

12056-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

12193-399: 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

12330-569: 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. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

12467-493: The distribution they were sampled from), whereas those based on the Pearson product-moment correlation coefficient are parametric tests since it is computed directly from the data values and thus estimates the parameter known as the population correlation . In probability theory , one may describe the distribution of a random variable as belonging to a family of probability distributions , distinguished from each other by

12604-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

12741-441: The extension of the word "parameter" to this sense, since it is not closely related to its mathematical sense, but it remains common. The term is also common in music production, as the functions of audio processing units (such as the attack, release, ratio, threshold, and other variables on a compressor) are defined by parameters specific to the type of unit (compressor, equalizer, delay, etc.). Mathematics Mathematics

12878-529: The first argument as a parameter, so as to create a family of functions f x : Y → Z . {\displaystyle f_{x}:Y\to Z.} The family is arranged so that for each object x {\displaystyle x} in X , {\displaystyle X,} there is exactly one function f x . {\displaystyle f_{x}.} In this example, curry {\displaystyle {\mbox{curry}}} itself becomes

13015-567: 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. 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",

13152-435: The function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general quadratic function by declaring Here, the variable x designates the function's argument, but a , b , and c are parameters (in this instance, also called coefficients ) that determine which particular quadratic function

13289-470: The functor B ↦ B × C {\displaystyle B\mapsto B\times C} is left adjoint to the functor A ↦ A C {\displaystyle A\mapsto A^{C}} . In the category of sets , the object Y X {\displaystyle Y^{X}} is called the exponential object . In the theory of function spaces , such as in functional analysis or homotopy theory , one

13426-418: The general idea of a type system in computer science is formalized into a specific algebra of types. For example, when writing f : X → Y {\displaystyle f\colon X\to Y} , the intent is that X {\displaystyle X} and Y {\displaystyle Y} are types , while the arrow → {\displaystyle \to }

13563-416: The independent variable, the position of the gas pedal. [Kilpatrick quoting Woods] "Now ... the engineers ... change the lever arms of the linkage ... the speed of the car ... will still depend on the pedal position ... but in a ... different manner . You have changed a parameter" In the context of a mathematical model , such as a probability distribution , the distinction between variables and parameters

13700-444: The integral, t is held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t , we then consider t to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration ). In statistics and econometrics , the probability framework above still holds, but attention shifts to estimating

13837-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , 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. Before

13974-400: The internal hom functor in the closed monoidal category. For the category of sets , the two are the same. When the product is the cartesian product, then the internal hom B ⇒ C {\displaystyle B\Rightarrow C} becomes the exponential object C B {\displaystyle C^{B}} . Currying can break down in one of two ways. One is if

14111-400: 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

14248-411: The main one by means of currying . Sometimes it is useful to consider all functions with certain parameters as parametric family , i.e. as an indexed family of functions. Examples from probability theory are given further below . W.M. Woods ... a mathematician ... writes ... "... a variable is one of the many things a parameter is not." ... The dependent variable, the speed of the car, depends on

14385-409: 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

14522-486: The mathematical context, the principle can be traced back to work in 1893 by Frege . The originator of the word "currying" is not clear. David Turner says the word was coined by Christopher Strachey in his 1967 lecture notes Fundamental Concepts in Programming Languages , but that source introduces the concept as "a device originated by Schönfinkel", and the term "currying" is not used, while Curry

14659-413: The mean μ and the variance σ². In these above examples, the distributions of the random variables are completely specified by the type of distribution, i.e. Poisson or normal, and the parameter values, i.e. mean and variance. In such a case, we have a parameterized distribution. It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for

14796-519: The movement of an object on the surface of a sphere much larger than the object (e.g. the Earth), there are two commonly used parametrizations of its position: angular coordinates (like latitude/longitude), which neatly describe large movements along circles on the sphere, and directional distance from a known point (e.g. "10km NNW of Toronto" or equivalently "8km due North, and then 6km due West, from Toronto" ), which are often simpler for movement confined to

14933-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. 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 . These problems and debates led to

15070-536: 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

15207-625: The pair f {\displaystyle f} and x {\displaystyle x} together as arguments, and returns f x . {\displaystyle f_{x}.} Using the same notation as above, partial application has the signature apply : ( [ ( X × Y ) → Z ] × X ) → [ Y → Z ] . {\displaystyle {\mbox{apply}}:([(X\times Y)\to Z]\times X)\to [Y\to Z].} Written this way, application can be seen to be adjoint to currying. The currying of

15344-496: The parameters of a distribution based on observed data, or testing hypotheses about them. In frequentist estimation parameters are considered "fixed but unknown", whereas in Bayesian estimation they are treated as random variables, and their uncertainty is described as a distribution. In estimation theory of statistics, "statistic" or estimator refers to samples, whereas "parameter" or estimand refers to populations, where

15481-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . 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. Evidence for more complex mathematics does not appear until around 3000  BC , when

15618-405: The physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal." The term can also be used in engineering contexts, however, as it is typically used in

15755-405: The physical sciences. In environmental science and particularly in chemistry and microbiology , a parameter is used to describe a discrete chemical or microbiological entity that can be assigned a value: commonly a concentration, but may also be a logical entity (present or absent), a statistical result such as a 95 percentile value or in some cases a subjective value. Within linguistics,

15892-405: The population from which the sample was drawn. Similarly, the sample variance (estimator), denoted S , can be used to estimate the variance parameter (estimand), denoted σ , of the population from which the sample was drawn. (Note that the sample standard deviation ( S ) is not an unbiased estimate of the population standard deviation ( σ ): see Unbiased estimation of standard deviation .) It

16029-658: The proof of numerous theorems. 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 . 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. Mathematics has since been greatly extended, and there has been

16166-444: The samples are taken from. A statistic is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of the population from which the sample was drawn. For example, the sample mean (estimator), denoted X ¯ {\displaystyle {\overline {X}}} , can be used as an estimate of the mean parameter (estimand), denoted μ , of

16303-446: The set of functions Y → Z . {\displaystyle Y\to Z.} Thus, something that maps x {\displaystyle x} to f x {\displaystyle f_{x}} will be of the type X → [ Y → Z ] . {\displaystyle X\to [Y\to Z].} With this notation, curry {\displaystyle {\mbox{curry}}}

16440-411: The set of functions from C {\displaystyle C} to A {\displaystyle A} . In symbols: Indeed, it is this natural bijection that justifies the exponential notation for the set of functions. As is the case in all instances of currying, the formula above describes an adjoint pair of functors : for every fixed set C {\displaystyle C} ,

16577-408: The space Y {\displaystyle Y} is locally compact Hausdorff , then is a homeomorphism . This is also the case when X {\displaystyle X} , Y {\displaystyle Y} and Y X {\displaystyle Y^{X}} are compactly generated , although there are more cases. One useful corollary is that a function

16714-454: The strict notion of currying and uncurrying is in the closed monoidal categories , which underpins a vast generalization of the Curry–Howard correspondence of proofs and programs to a correspondence with many other structures, including quantum mechanics, cobordisms and string theory. The concept of currying was introduced by Gottlob Frege , developed by Moses Schönfinkel , and further developed by Haskell Curry . Uncurrying

16851-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

16988-568: The study of approximation and discretization with special focus on rounding errors . 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 . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

17125-409: The subset of continuous functions. Here, curry {\displaystyle {\text{curry}}} is the bijection while uncurrying is the inverse map. If the set Y X {\displaystyle Y^{X}} of continuous functions from X {\displaystyle X} to Y {\displaystyle Y} is given the compact-open topology , and if

17262-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

17399-487: The title of his main treatise . 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. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

17536-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. 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

17673-468: The type Y {\displaystyle Y} , and the function itself returns the type Z {\displaystyle Z} . The curried form of f is defined as where λ {\displaystyle \lambda } is the abstractor of lambda calculus. Since curry takes, as input, functions with the type ( X × Y ) → Z {\displaystyle (X\times Y)\to Z} , one concludes that

17810-482: The type of curry itself is The → operator is often considered right-associative , so the curried function type X → ( Y → Z ) {\displaystyle X\to (Y\to Z)} is often written as X → Y → Z {\displaystyle X\to Y\to Z} . Conversely, function application is considered to be left-associative , so that f ( x , y ) {\displaystyle f(x,y)}

17947-444: The values of a finite number of parameters . For example, one talks about "a Poisson distribution with mean value λ". The function defining the distribution (the probability mass function ) is: This example nicely illustrates the distinction between constants, parameters, and variables. e is Euler's number , a fundamental mathematical constant . The parameter λ is the mean number of observations of some phenomenon in question,

18084-786: The word "parameter" is almost exclusively used to denote a binary switch in a Universal Grammar within a Principles and Parameters framework. In logic , the parameters passed to (or operated on by) an open predicate are called parameters by some authors (e.g., Prawitz , "Natural Deduction"; Paulson , "Designing a theorem prover"). Parameters locally defined within the predicate are called variables . This extra distinction pays off when defining substitution (without this distinction special provision must be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables , and when defining substitution have to distinguish between free variables and bound variables . In music theory,

18221-403: Was described by Bard as follows: In analytic geometry , a curve can be described as the image of a function whose argument, typically called the parameter , lies in a real interval . For example, the unit circle can be specified in the following two ways: with parameter t ∈ [ 0 , 2 π ) . {\displaystyle t\in [0,2\pi ).} As

18358-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . 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 . Euclidean geometry was developed without change of methods or scope until

18495-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

18632-437: Was not specifically studied by mathematicians. 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 but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

18769-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves 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. In

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