Computer scientists have used at least three notations for monads, and terminology varies. The following descriptions use the Haskell names for operations. Monads in functional programming - Category-theoretic notation. Following the definition of a monad in category theory, a program can specify a monad as a set of three type-generic functions: A function return that turns a value in any type to a value into a corresponding monadic type; A function fmap that ...

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Monads in functional programming, Monads in functional programming - Example: Maybe monad, Monads in functional programming - Notations, Monads in functional programming - Category-theoretic notation, Monads in functional programming - Bind operator, Monads in functional programming - do-notation, Monads in functional programming - Monad types, Monads in functional programming - More examples, Monads in functional programming - Maybe monad revisited, Monads in functional programming - Identity monad, Monads in functional programming - Lists, Monads in functional programming - State transformers, Monads in functional programming - I/O, Monads in functional programming - Others

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Monas Monadum In Leibniz' system of monads, the supreme monad, which is infinite and upon which there depend three classes of finite monads. This supreme monad held the place of God, an infinite perfect spirit, a Person of absolute power, wisdom, and goodness. I

n this case, the supreme monad is cosmically more than a person -- for etymologically person means a mask or vehicle through and from which issue the attributes and powers of something incomparably higher than itself. Equivalent to the summit of the human hierarchy.

(See also: Monas Monadum, Mysticism, Mysticism Dictionary)

A Theosophical definition of Human Monad :

Human Monad

In theosophical terminology the human monad is that part of man's constitution which is the root of the human ego. After death it allies itself with the upper duad, atma-buddhi, and its inclusion within the bosom of the upper duad produces the source whence issues the Reincarnating Ego at its next rebirth. The monad per se is an upper duad alone, but the attributive adjective "human" is given to it on account of the reincarnating ego which it contains within itself after death. This last usage is rather popular and convenient than strictly accurate.

See also: Human Monad, Mysticism, Body Mind and Soul)

Human Soul The clothing or ray of the human ego; it is the egoic center in manas, under the influence of both buddhi and the kamic nature.

We may speak of a threefold human soul -- buddhi-manas or the spiritual soul, manas or the human soul, and kama-manas or the animal soul, each the expression of its own ego. Each ego is the expression of its monad. The characteristic of the human soul is duality, affording the field for the interaction of spiritual and lower forces.

(See also: Human Monad, Mysticism, Mysticism Dictionary, Occultism, Occultism Dictionary)

Human Monad In the human constitution, the fourth monadic focus or center on the descending scale of individualizing consciousness. It is the basis or root of the human ego from which emanates the human soul -- a temporary or periodic appearance enduring for one incarnation, having for its range of consciousness the ordinary human consciousness of daily life.

At death the essence of the human soul is united to the human ego, which in its turn at the second death is reunited with the upper duad (atma-buddhi); and the human ego thereupon enters into the state of consciousness called devachan.

Having become at one with its spiritual parent, at least for the duration of devachan, the ego rests and digests its garnered store of wisdom, knowledge, and experience, and upon the completion of this period of devachanic recuperation it issues forth again when the karmic hour strikes, once more to become the human ego at its succeeding birth.

(See also: Human Monad, Mysticism, Mysticism Dictionary, Occultism, Occultism Dictionary)

The categorical dual definition is a formal definition of a comonad; this can be said quickly in the terms that a comonad for a category C is a monad for the opposite category Cop. It is therefore a functor U from C to itself, with a set of axioms for counit and comultiplication that come from reversing the arrows everywhere in the definition just given. Since a comonoid is not a basic structure in abstract algebra, this is less familiar at a ...

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Monad category theory, Monad category theory - Introduction, Monad category theory - Formal definition, Monad category theory - Comonads and their importance, Monad category theory - Examples, Monad category theory - Uses

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If C is a category, a monad on C consists of a functor T : C → C together with two natural transformations: η : 1C → T (where 1C denotes the identity functor on C) and μ : T2 → T (where T2 is the functor T o T from C to C). These are required to fulfill the following axioms: μ o Tμ = μ o μT (as natural transformations T3See also:

Monad category theory, Monad category theory - Introduction, Monad category theory - Formal definition, Monad category theory - Comonads and their importance, Monad category theory - Examples, Monad category theory - Uses

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Astral Monad or Soul The animal soul or vital-astral soul, the lowest and feeblest reflection or vehicle of the divine monad; when enlightened by the human monad, it produces the human being known today.

(See also: Astral Monad, Soul, Mysticism, Mysticism Dictionary, Occultism, Occultism Dictionary)

Monads in functional programming - Maybe monad revisited. Now a complete description of the Maybe monad is possible: data Maybe a = Just a | Nothing return x = Just x fmap f (Just x) = Just (f x) fmap f Nothing = Nothing (Just x) >>= f = f x Nothing >>= f = Nothing The division example is then implemented as follows: x // y = do a <- x; b <- y; when (b == 0) (fail "0 divisor") ; return (a / b) x ++ y = do a <- x; b <- y; return (a + b) ...

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Monads in functional programming, Monads in functional programming - Example: Maybe monad, Monads in functional programming - Notations, Monads in functional programming - Category-theoretic notation, Monads in functional programming - Bind operator, Monads in functional programming - do-notation, Monads in functional programming - Monad types, Monads in functional programming - More examples, Monads in functional programming - Maybe monad revisited, Monads in functional programming - Identity monad, Monads in functional programming - Lists, Monads in functional programming - State transformers, Monads in functional programming - I/O, Monads in functional programming - Others

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Elements of an interior algebra satisfying the condition xI = x are called open. The complements of open elements are called closed and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements which are both ...

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Interior algebra, Interior algebra - Open and closed elements, Interior algebra - Morphisms of interior algebras, Interior algebra - Homomorphisms, Interior algebra - Topomorphisms, Interior algebra - Relationships to other areas of mathematics, Interior algebra - Topology, Interior algebra - Modal logic, Interior algebra - Preorders, Interior algebra - Monadic Boolean algebras, Interior algebra - Heyting algebras, Interior algebra - Derivative algebras

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Interior algebra - Homomorphisms. Since interior algebras are algebraic structures we can speak of interior algebra homomorphisms. Given two interior algebras A and B, a map f : A → B is an interior algebra homomorphism if and only if it is a homomorphism between the underlying Boolean algebras of A and B and in addition preserves interiors (and hence equivalently, preserves closures) i.e. f(xI) = f(x)I ...

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Interior algebra, Interior algebra - Open and closed elements, Interior algebra - Morphisms of interior algebras, Interior algebra - Homomorphisms, Interior algebra - Topomorphisms, Interior algebra - Relationships to other areas of mathematics, Interior algebra - Topology, Interior algebra - Modal logic, Interior algebra - Preorders, Interior algebra - Monadic Boolean algebras, Interior algebra - Heyting algebras, Interior algebra - Derivative algebras

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Purely functional programs have no side-effects. Since functions do not modify state, no data may be changed by parallel function calls. For this reason, pure functions are always thread-safe, a fact which is exploited by languages that use call-by-future evaluation. Because ordering of side-effects does not have to be preserved in their absence, some languages (such as Haskell) use call-by-need evaluation for pure functions. "Pure" functional programming languages typically enforce referential transparency, which is the notion ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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Functional languages have long been criticised as resource-hungry, both in terms of CPU resources and memory. This was mainly due to two factors: some early functional languages were implemented with little concern for efficiency non-functional languages achieved speed in part by leaving out features such as bounds checking and garbage collection which are considered by many to be important parts of modern computing fr ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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Lambda calculus, invented by Alonzo Church in the 1930s, provides a theoretical framework for describing functions and their evaluation. Though it is a mathematical abstraction rather than a programming language, lambda calculus forms the basis of almost all functional programming languages today. The first computer-based functional programming language was Information Processing Language (IPL), developed by Newell, Shaw, and Simon at RAND Corporation for the JOHNNIAC computer in the mid-1950s. A much-improved functional programming l ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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A powerful mechanism used in functional programming is the notion of higher-order functions. Functions are higher-order when they can take other functions as arguments, and/or return functions as results. (The differential operator in calculus is a common example of a function that maps a function to a function.) Higher-order functions are closely related to first-class functions, in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two i ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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Functional programming can be contrasted with imperative programming. Functional programming appears to be missing several constructs often (though incorrectly) considered essential to an imperative language such as C or Pascal. For example, in strict functional programming, there is no explicit memory allocation and no explicit variable assignment. However, these operations occur automatically when a function is invoked: memory allocation occurs to create space for the parameters and the return value, and assignment occurs to copy the param ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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It is very difficult to grasp Leibniz's philosophical thinking, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He only wrote two philosophical treatises, and the only one he published in his lifetime, the Théodicée of 1710, is as much theological as philosophical. Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a run ...

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Gottfried Leibniz, Gottfried Leibniz - Life, Gottfried Leibniz - Coming of age, Gottfried Leibniz - Career, Gottfried Leibniz - Writings, Gottfried Leibniz - Posthumous reputation, Gottfried Leibniz - Philosopher, Gottfried Leibniz - The Principles, Gottfried Leibniz - The Monads, Gottfried Leibniz - Theodicy and optimism, Gottfried Leibniz - Symbolic thought, Gottfried Leibniz - Formal logic, Gottfried Leibniz - Mathematician, Gottfried Leibniz - The calculus, Gottfried Leibniz - Topology, Gottfried Leibniz - Scientist and engineer, Gottfried Leibniz - Physics, Gottfried Leibniz - Other natural science, Gottfried Leibniz - Social science, Gottfried Leibniz - Technology, Gottfried Leibniz - The librarian, Gottfried Leibniz - Advocate of scientific societies, Gottfried Leibniz - Lawyer Moralist Theologian, Gottfried Leibniz - Ecumenism, Gottfried Leibniz - Philologist, Gottfried Leibniz - Sinophile, Gottfried Leibniz - Universal Genius, Gottfried Leibniz - Works, Gottfried Leibniz - Secondary literature, Gottfried Leibniz - Other works cited, Gottfried Leibniz - Quotes

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The first computer-based functional programming language was Information Processing Language (IPL) from the RAND corporation. Another very old functional language is Lisp, though neither the original LISP nor modern Lisps such as Common Lisp are pure-functional. Some Lisp variants include Scheme, Dylan, and Logo (though Logo is an imperitive language). The modern canonical examples are Haskell and members of the ML family including SML and OCaml. Others include Erlang, Clean, and Miranda. A third type of a commonly used functional language is Xslt. Ano ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

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Leibniz was perhaps the first major European intellect to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other work by, European Christian missionaries posted in China. He concluded that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 0 to 111111, and mistakenly concluded that this mapping was evidence of major Chinese accomplishments in th ...

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Gottfried Leibniz, Gottfried Leibniz - Life, Gottfried Leibniz - Coming of age, Gottfried Leibniz - Career, Gottfried Leibniz - Writings, Gottfried Leibniz - Posthumous reputation, Gottfried Leibniz - Philosopher, Gottfried Leibniz - The Principles, Gottfried Leibniz - The Monads, Gottfried Leibniz - Theodicy and optimism, Gottfried Leibniz - Symbolic thought, Gottfried Leibniz - Formal logic, Gottfried Leibniz - Mathematician, Gottfried Leibniz - The calculus, Gottfried Leibniz - Topology, Gottfried Leibniz - Scientist and engineer, Gottfried Leibniz - Physics, Gottfried Leibniz - Other natural science, Gottfried Leibniz - Social science, Gottfried Leibniz - Technology, Gottfried Leibniz - The librarian, Gottfried Leibniz - Advocate of scientific societies, Gottfried Leibniz - Lawyer Moralist Theologian, Gottfried Leibniz - Ecumenism, Gottfried Leibniz - Philologist, Gottfried Leibniz - Sinophile, Gottfried Leibniz - Universal Genius, Gottfried Leibniz - Works, Gottfried Leibniz - Secondary literature, Gottfried Leibniz - Other works cited, Gottfried Leibniz - Quotes

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No philosopher has ever had as much experience with practical affairs of state as Leibniz, Marcus Aurelius possibly excepted. Leibniz's writings on law, ethics, and politics (e.g., AG 19, 94, 111, 193; Riley 1988; LL §§2, 7, 20, 29, 44, 59, 62, 65; W I.1, IV.1-3) were long overlooked by English speaking scholars but this has changed of late; see (in order of difficulty) Jolley (2005: chpt. 7), Gregory Bro ...

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Gottfried Leibniz, Gottfried Leibniz - Life, Gottfried Leibniz - Coming of age, Gottfried Leibniz - Career, Gottfried Leibniz - Writings, Gottfried Leibniz - Posthumous reputation, Gottfried Leibniz - Philosopher, Gottfried Leibniz - The Principles, Gottfried Leibniz - The Monads, Gottfried Leibniz - Theodicy and optimism, Gottfried Leibniz - Symbolic thought, Gottfried Leibniz - Formal logic, Gottfried Leibniz - Mathematician, Gottfried Leibniz - The calculus, Gottfried Leibniz - Topology, Gottfried Leibniz - Scientist and engineer, Gottfried Leibniz - Physics, Gottfried Leibniz - Other natural science, Gottfried Leibniz - Social science, Gottfried Leibniz - Technology, Gottfried Leibniz - The librarian, Gottfried Leibniz - Advocate of scientific societies, Gottfried Leibniz - Lawyer Moralist Theologian, Gottfried Leibniz - Ecumenism, Gottfried Leibniz - Philologist, Gottfried Leibniz - Sinophile, Gottfried Leibniz - Universal Genius, Gottfried Leibniz - Works, Gottfried Leibniz - Secondary literature, Gottfried Leibniz - Other works cited, Gottfried Leibniz - Quotes

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The following episode from the life of Leibniz illustrates the breadth of his genius, and the difficulties awaiting those who try to come to terms with it. While making his grand tour of European archives to research the Brunswick family history he never completed, Leibniz stopped in Wien, May 1688 – February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains. His proposal that the str ...

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Gottfried Leibniz, Gottfried Leibniz - Life, Gottfried Leibniz - Coming of age, Gottfried Leibniz - Career, Gottfried Leibniz - Writings, Gottfried Leibniz - Posthumous reputation, Gottfried Leibniz - Philosopher, Gottfried Leibniz - The Principles, Gottfried Leibniz - The Monads, Gottfried Leibniz - Theodicy and optimism, Gottfried Leibniz - Symbolic thought, Gottfried Leibniz - Formal logic, Gottfried Leibniz - Mathematician, Gottfried Leibniz - The calculus, Gottfried Leibniz - Topology, Gottfried Leibniz - Scientist and engineer, Gottfried Leibniz - Physics, Gottfried Leibniz - Other natural science, Gottfried Leibniz - Social science, Gottfried Leibniz - Technology, Gottfried Leibniz - The librarian, Gottfried Leibniz - Advocate of scientific societies, Gottfried Leibniz - Lawyer Moralist Theologian, Gottfried Leibniz - Ecumenism, Gottfried Leibniz - Philologist, Gottfried Leibniz - Sinophile, Gottfried Leibniz - Universal Genius, Gottfried Leibniz - Works, Gottfried Leibniz - Secondary literature, Gottfried Leibniz - Other works cited, Gottfried Leibniz - Quotes

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