The word monad comes from the Greek word μονάς (from the word μόνος, which means "one", "single", "unique") and has had many meanings in different contexts in philosophy, mathematics, computing and music: Among the Pythagoreans (followers of Pythagoras) the monad was the first thing that came into existence. The monad begot the dyad, which begot the numbers, the numbers begat points, which begot lines, which begat two-dimensional entities, which begat three-dimensional entities, which begat bodies, ...

Read more here: » Monad: Encyclopedia - Monad

Monad:

The One created by the Logos. At the Monadic level of consciousness one is fully aware of the oneness of all things, but is not yet receiving Logoic light directly.

(See also: Monad, Body Mind and Soul)

monad

An indivisible and divine spiritual life-atom. The immortal self within each person that lives on in successive incarnations, progressing to the stature of the adept beyond which extends unlimited evolution

(See also: Monad, Body Mind and Soul)

In many Gnostic systems (and heresiologies), God is known as the Monad, the One, The Absolute Aion teleos (The Perfect Æon), Bythos (Depth or Profundity, Βυθος), Proarkhe (Before the Beginning, προαρχη), and E Arkhe (The Beginning, η αρχη). God is the high source of the pleroma, the region of light. The various emanations of God are called æons. Within certain variations of Gnosticism, especially those inspired by Monoimus, the Monad was the highest God w ...

Read more here: » Monad Gnosticism: Encyclopedia - Monad Gnosticism

A Theosophical definition of Monad :

Monad

A spiritual entity which to us humans is indivisible; it is a divine-spiritual life-atom, but indivisible because its essential characteristic, as we humans conceive it, is homogeneity; while that of the physical atom, above which our consciousness soars, is divisible, is a composite heterogeneous particle.

Monads are eternal, unitary, individual life-centers, conscious-ness-centers, deathless during any solar manvantara, therefore ageless, unborn, undying. Consequently, each one such  - and their number is infinite  - is the center of the All, for the divine or the All is THAT which has its center everywhere, and its circumference or limiting boundary nowhere.

Monads are spiritual-substantial entities, self-motivated, self-impelled, self-conscious, in infinitely varying degrees, the ultimate elements of the universe. These monads engender other monads as one seed will produce multitudes of other seeds; so up from each such monad springs a host of living entities in the course of illimitable time, each such monad being the fountainhead or parent, in which all others are involved, and from which they spring.

Every monad is a seed, wherein the sum total of powers appertaining to its divine origin are latent, that is to say unmanifested; and evolution consists in the growth and development of all these seeds or children monads, whereby the universal life expresses itself in innumerable beings.

As the monad descends into matter, or rather as its ray  - one of other innumerable rays proceeding from it  - is propelled into matter, it secretes from itself and then excretes on each one of the seven planes through which it passes, its various vehicles, all overshadowed by the self, the same self in you and in me, in plants and in animals, in fact in all that is and belongs to that hierarchy. This is the one self, the supreme self or paramatman of the hierarchy. It illumines and follows each individual monad and all the latter's hosts of rays  - or children monads. Each such monad is a spiritual seed from the previous manvantara, which manifests as a monad in this manvantara; and this monad through its rays throws out from itself by secretion and then excretion all its vehicles. These vehicles are, first, the spiritual ego, the reflection or copy in miniature of the monad itself, but individualized through the manvantaric evolution, "bearing" or "carrying" as a vehicle the monadic ray. The latter cannot directly contact the lower planes, because it is of the monadic essence itself, the latter a still higher ray of the infinite Boundless composed of infinite multiplicity in unity. (See also Individuality)

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

Monad (Ancient Greek). The Unity, the one ; but in Occultism it often means the unified triad, Atma-Buddhi-Manas, or the duad, Atma-Buddhi, that immortal part of man which reincarnates in the lower kingdoms, and gradually progresses through them to Man and then to the final goal -  Nirvana.

(See also: Monad, Theosophy, Spirituality, Body mind and Soul, Spiritual Dictionary, )

Monads are part of category theory. A monad itself is just a data type. It obeys three properties: It must have the ability to contain another data type. A function ("type constructor") must be available, which takes the 'contents' and produces an instance of the monad containing the contents. A function ("bind") is required that takes: A monad containing type A A function that converts from a type A, to a monad of type B and returns a mo ...

See also:

Monads, Monads - Monads in applied computer science, Monads - Theory

Read more here: » Monads: Encyclopedia II - Monads - Theory

Traditionally, computer programs have been written in a procedural fashion, with a list of statements which the machine executes one after the other. However, several other types of programming exist - one of which is functional programming. In functional programming, the entire program is expressed as a series of mathematical functions. No order of execution is specified. The interpreter (or compiler) looks at the interrelationships between ...

See also:

Monads, Monads - Monads in applied computer science, Monads - Theory

Read more here: » Monads: Encyclopedia II - Monads - Monads in applied computer science

Any type can be made an instance of the Monad type class by defining the method return for making such a monad and the operator >>= for sequencing a monad operation to another. For the definitions to make sense, they should follow the monad laws. Optionally, an error method and a result-discarding bind can be specified. Simple types with useful monad interpretations include a list and Either. List can be used to produce results of an operation sequence for all combinations of inputs (ie. simulate nondet ...

See also:

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

Read more here: » Monads in functional programming: Encyclopedia II - Monads in functional programming - Monad types

MONAD/MONAS

Any entity acting as "one": the self, earth, universe, a society, even the life force at the atomic level. HPB (The Secret Doctrine) pointed out that "each atom is doomed to incessant differentiation."

(See also: MONAD, MONAS, Magick, Mysticism, Mysticism Dictionary, Body Mind and Soul, )

Monad, Monas [from Greek monas a unit, individual, atom]

A unit, a one; something nondivisible and which is therefore conceived of as real, in contradistinction to compound things which (as compounds) are not real.

In the Pythagorean system the Duad emanates from the higher and solitary Monas, which is thus the First Cause or First Logos, the Duad being the Second Cause or Logos; and from the second emanates the third stage of individuality, the Triad, Third Cause or Logos. In the human constitution the Monas signifies atman, the Duad buddhi, and the Triad signifies manas.

The term monad was adopted from Greek philosophy by Bruno, Leibniz, and others. According to Leibniz there can be but one ultimate cosmic reality or monad, the universe; but he recognizes an innumerable multiplicity of monads which pervade the universe, copies or reflections of the universal monad regarded as real except in their relation to the universal monad. He divides his derivative monads into three classes: rational souls; sentient but irrational monads; and material monads, or organic and inorganic bodies.

As regards the material monads, while recognizing that corporeal matter is compound, and the attributes by which we perceive it unreal, unlike Berkeley, he does not deny its existence but regards it essentially as monadic. Thus his universe is an aggregate of individuals. The relations of these individuals to each other and to the universal is a supreme harmony, implying both individuality and coordination, thus reconciling the antinomy of bonds of law and freedom. The interrelations of various groups of monads is as a series of hierarchies.

Theosophical usage is largely the same as that of Leibniz, as the focus or heart in any individual being, of all its divine, spiritual, and intellectual powers and attributes -- the immortal part of its being. In The Secret Doctrine we find a triadic union of gods-monads-atoms, related to each other as spirit-soul-body (or more accurately spirit, spirit-soul, and spirit-soul-body). Monads and atoms are related to each other as the energic and the material side of manifestation, the atoms being the reflections, veils, or projections of and from the monads themselves.

Monads are the ultimate elements of the universe, spiritual-substantial entities, self-motivated, self-impelled, self-conscious, in infinitely varying degrees. They engender other monads, which in turn engender others, and thus springs up the host of living entities forming the immense variety and unity of the manifested world. As any monad descends into matter, it secretes from itself various veils or vehicles adapted for its self-expression on the various cosmic planes. Thus in man there is the divine monad, the spiritual monad, the higher human or chain monad, the lower human or globe monad, the animal monad, and the astral-physical monad. The following diagram shows the relations between the cosmic principles; the monads, egos and souls in the human being; and the human principles

The monad, as its name implies, is ever-enduring as an individual, although at the end of each manvantara it rises into a still higher or divine stage of perfect union with the boundless divine, only to re-issue forth again in due course as the monad it was before, thus beginning a new, immensely long time period of active individualized life as a spiritual consciousness-center. Thus it is that even the monads evolve, each on its own plane, for the hierarchies of the monads are innumerable and exist in all-various degrees at stages of evolutionary progression on the endless ladder of cosmic life.

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

Monadic Ray The monad, that divine-spiritual-intellectual seed or originant of each evolving being, does not itself descend into the planes of matter, but shoots forth from itself a multitude of rays.

Each such rays forms the essential nature of the complex evolving being to which it pertains, and hence the monad is the primal or ultimate source of all that being's life and characteristic attributes, the immortal part of the being, whether that being be human, animal, vegetable, mineral, or what not. In man it is his essential self; it persists throughout all the evolutionary transformations in the life cycle and gathers around itself the life-atoms at each new incarnation of the reincarnating ego.

Thus the monad in any person is his inner god, the celestial buddha of his own septenary constitution, or again his individual Immanent Christ.

The rays from the person's individual monad which form the complex essential nature of his being, are the sources of the different centers in the human constitution, and in themselves are children monads, as it were, from their common source.

(See also: Monadic Ray, Mysticism, Mysticism Dictionary)

If F and G are an adjoint pair of functors, with F left adjoint to G, then the composition G o F will be a monad. Note that therefore a monad is a functor from a category to itself; and that if F and G were actually inverses as functors the corresponding monad would be the identity functor. In general adjunctions are not equivalences — they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The othe ...

See 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

Read more here: » Monad category theory: Encyclopedia II - Monad category theory - Introduction

Animal Monad

During its evolution in the animal kingdom the monad is called animal monad. The vegetable monads become animal monads through acquiring emotional consciousness.

(See also: Animal Monad, New Age Spirituality, Body Mind and Soul)

Before discussing the details of monads, it might help to give a motivating example. Consider a function that is undefined for some known values, such as division. Division might occur repeatedly in a calculation, like this one: par:: Real -> Real -> Real -- Takes two real numbers and returns another par x y = 1 / ((1 / x) + (1 / y)) Instead of avoiding any errors by checking whether each divisor is zero, it might be convenient to have a modified division ...

See also:

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

Read more here: » Monads in functional programming: Encyclopedia II - Monads in functional programming - Example: Maybe monad

Monadic Envelope.

See AURIC EGG

(See also: Monadic Envelope, Mysticism, Mysticism Dictionary)

The basis of most if not all medieval European music theory is the hexachordal system. This system attempted to explain the pitch relationships in the music of the time by grouping pitches into sets of six, or hexachords. In each hexachord, all adjacent pitches are a whole tone apart, except for the middle two, which are separated by a half tone. In each hexachord, these pitches are called ut, re, mi, fa, sol, and la, so the half tone is always between mi and fa. Note that most pitches have several s ...

See also:

Hexachord, Hexachord - Middle Ages, Hexachord - Set theory

Read more here: » Hexachord: Encyclopedia II - Hexachord - Middle Ages

Peregrinations of the Monad Used mainly for the post-mortem states and conditions of the spiritual monad plus its movements in and through the solar system guided by certain dominating spiritual-psychological factors, both in the monad itself and in the solar system.

See also INNER ROUNDS; OUTER ROUNDS

(See also: Peregrinations of the Monad, Mysticism, Mysticism Dictionary, Occultism, Occultism Dictionary)

Spiritual Monad The second monadic center in the descending scale of intimately related human monadic centers; in the septenary constitution, atma-buddhi-manas, with an emphasis on the buddhi-manas, atman being the divine monad. It is man's individual monad, the spiritual center of his own stream of consciousness, in the heart of which abides his inner god or "Father in heaven."

"The human constitution is a composite or compound, and may be figurated . . . as a stream of consciousness flowing forth from the deathless Center or Spiritual Monad, which last is at once the immortal Root of the human being and his Essential Self" (ET 718). It corresponds to the spiritual self or jivatman.

After death, when the second death occurs, man's consciousness is withdrawn from the higher astral regions into the next superior sphere or plane -- the human monad is indrawn into the spiritual monad. Then occurs the state of devachan.

(See also: Spiritual Monad, Mysticism, Mysticism Dictionary, Body mind and Soul)

The most important examples to think about when hearing the term "monad" are the free group example mentioned above, and closure operators. Here's another example on the category Set: for a set A let T(A) be the power set of A and for a function f : A → B let T(f) be the function between the power sets induced by taking direct images under f. For every set A, we have a map ηA : A → T(A), which assigns to e ...

See 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

Read more here: » Monad category theory: Encyclopedia II - Monad category theory - Examples