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see Bruno, GiordanoBruno, Giordano
, 1548–1600, Italian philosopher, b. Nola. The son of a professional soldier, he entered the Dominican order early in his youth and was ordained a priest in 1572, but he was accused of heresy and fled (c.1576) to take up a career of study and travel.
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; Leibniz, Gottfried Wilhelm, Baron vonLeibniz or Leibnitz, Gottfried Wilhelm, Baron von
, 1646–1716, German philosopher and mathematician, b. Leipzig.
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a concept used in a number of philosophical systems to designate the constituents of existence. In classical philosophy this concept was introduced as the first, or all-explaining, principle by Pythagoreanism, which regarded number and proportion as the source of all things. From the Pythagoreans the concept of the monad was adopted by Plato (the dialogue Philebus) and by the Neoplatonists, who interpreted it pantheistically as the first principle that unfolds and reproduces itself in many things by means of emanation.

The concept entered modern philosophy in the pantheism of Nicholas of Cusa and G. Bruno. According to Bruno, monads mirror the infinite universe in accordance with the principle of the unity of the microcosm and the macrocosm. In the 17th century the concept of the monad was important in the philosophy of the Spanish scholastic F. Suárez, the English Platonist Henry More, and the German natural philosopher F. M. van Helmont. It was the pivotal concept for the entire philosophical system of G. W. von Leibniz, who developed the doctrine of monadology. According to his definition, a monad is an ultimate and simple (irreducible) active substance that has a spiritual nature and that apprehends and mirrors the entire world. There is an infinite number of monads, all of which coexist in a state of preestablished harmony. Because the spiritual nature of monads precludes their interaction, harmony among them is reduced to a divinely preestablished cosmic unity. Although it was a classical doctrine of objective idealism, Leibniz’ monadology played an important role in the spread of a dynamic, dialectical view of nature. It contained such ideas as the principle of the universal interrelation of things, the principle of the uniformity of natural laws, the law of conservation, and the concept of universal variability and self-development.

After Leibniz the concept of monads was elaborated in the spirit of idealistic rationalism by the followers of C. von Wolff. In the 19th century the ideas of monadology were echoed in the views of a number of German philosophers, including J. Herbart and R. Lotze, and in the 20th century, in the philosophy of E. Husserl (Germany), A. Whitehead (Great Britain), and R. Honigswald (Germany-USA). The monadological approach is the foundation for the philosophical views of many of the representatives of personalism, including C. Renouvier, H. Carr, and J. McTaggart.


Lenin, V. I. Poln. sobr. soch., 5th ed., vol. 29, pp. 67–76.
Cramer, W. Die Monade: Das philosophische Problem von Ursprung. Stuttgart, 1954.
Heimsoeth, H. Atom, Seele, Monade.…. Mainz, 1960.
Horn, J. C. Monade und Begriff. Wiesbaden-Munich, 1965.



1. Philosophy
a. any fundamental singular metaphysical entity, esp if autonomous
b. (in the metaphysics of Leibnitz) a simple indestructible nonspatial element regarded as the unit of which reality consists
c. (in the pantheistic philosophy of Giordano Bruno) a fundamental metaphysical unit that is spatially extended and psychically aware
2. a single-celled organism, esp a flagellate protozoan
3. an atom, ion, or radical with a valency of one


(theory, functional programming)
/mo'nad/ A technique from category theory which has been adopted as a way of dealing with state in functional programming languages in such a way that the details of the state are hidden or abstracted out of code that merely passes it on unchanged.

A monad has three components: a means of augmenting an existing type, a means of creating a default value of this new type from a value of the original type, and a replacement for the basic application operator for the old type that works with the new type.

The alternative to passing state via a monad is to add an extra argument and return value to many functions which have no interest in that state. Monads can encapsulate state, side effects, exception handling, global data, etc. in a purely lazily functional way.

A monad can be expressed as the triple, (M, unitM, bindM) where M is a function on types and (using Haskell notation):

unitM :: a -> M a bindM :: M a -> (a -> M b) -> M b

I.e. unitM converts an ordinary value of type a in to monadic form and bindM applies a function to a monadic value after de-monadising it. E.g. a state transformer monad:

type S a = State -> (a, State) unitS a = \ s0 -> (a, s0) m `bindS` k = \ s0 -> let (a,s1) = m s0 in k a s1

Here unitS adds some initial state to an ordinary value and bindS applies function k to a value m. (`fun` is Haskell notation for using a function as an infix operator). Both m and k take a state as input and return a new state as part of their output. The construction

m `bindS` k

composes these two state transformers into one while also passing the value of m to k.

Monads are a powerful tool in functional programming. If a program is written using a monad to pass around a variable (like the state in the example above) then it is easy to change what is passed around simply by changing the monad. Only the parts of the program which deal directly with the quantity concerned need be altered, parts which merely pass it on unchanged will stay the same.

In functional programming, unitM is often called initM or returnM and bindM is called thenM. A third function, mapM is frequently defined in terms of then and return. This applies a given function to a list of monadic values, threading some variable (e.g. state) through the applications:

mapM :: (a -> M b) -> [a] -> M [b] mapM f [] = returnM [] mapM f (x:xs) = f x `thenM` ( \ x2 -> mapM f xs `thenM` ( \ xs2 -> returnM (x2 : xs2) ))
References in periodicals archive ?
Though not explicitly defined here, in the correspondence with Clarke, Leibniz addresses the immaterial, immanent force or entelechy within all living and non-living matter--the monad--as a spiritual "simple substance," or "soul" (A Collection of Papers, 245); each monad has its own organic body (distinct from the body of which it is the entelechy), which in turn is composed of monads, ad infinitum.
8) As far as it might appear from contemporary globalisation, a model of comparative literary study centred on Enlightenment philosophy, concentrating on the monad, provides a way of looking at contemporary literature as constituting a series of echoes or reverberations between individual texts, and of suggesting a way to consider each text as possessing the whole of literature, even as it is also a constituent part.
In the next section we recall some results about categorical tensors, focusing on how when tensors exist they induce a monad on the category V over which our ambient category C is enriched.
Here again Garber convincingly argues against those competing interpretations, most notably Robert Adams's 'qualified monad conception' (93-7), that try to assimilate these ideas to Leibniz's subsequent monadology by downplaying the significance of corporeal substance talk in these middle years and emphasizing the continuity of this metaphysics with Leibniz's latter thought.
His creation of the monad's capacity for internalising relations allows him to welcome Leibniz's treatment of space and time as dialectical whereas Leibniz accepts internal modifications but denies any effect from other monads, leaving little space for dialectics, either as method or in actuality.
Fragments, monads and vessels are finished with a simple clay-based glaze to seal and add surface patina over the slip.
Thus, while a finite monad neither exists "in" space nor has extension, it nevertheless represents the universe as if from a point of view "rather as the same town is differently represented according to the different situations of the person who looks at it.
Phases Number of Number of Abnormalities Number analyzed abnormal of cells cells cells Metaphase I 142 32 Non-oriented 12 chromosome Stickiness 20 Anaphase I 162 48 Laggards 38 Laggards and 10 precocious migration Telophase I 190 16 Micronuclei 16 Prophase II 176 12 Micronuclei 12 Metaphase II 216 24 precocious migration 02 Stickiness 22 Anaphase II 130 46 Laggards 18 Stickiness 28 Telophase II 152 16 Micronuclei 16 Tetrad with 10 micronuclei Triad 11 Triad with microcyte 18 Tetrad 204 82 Dyad 22 Dyad with microcyte 10 Monad 07 Monad with microcyte 04
This study confirmed the importance of Epping Forest (40% of occupied monads in the survey were in the Forest), a large (ca 2400 ha) and continuous area with a great diversity of habitats, for the conservation of O.
Deleuze's conceptualization reflects Leibniz's idea that "bifurcations and divergences of series are genuine borders between incompossible worlds, such that the monads that exist wholly include the compossible world that moves into existence" (1993, 81).
0] > 0) is the strength of a monad, m(t) is the mass of a particle at time t, Q is the strength of the particle, N is the number of monads that make up the particle, [rho] is the density of the [OMEGA](0) substratum, t [greater than or equal to] 0.
Other topics include Temperley-Lieb and non-crossing partition planar algebras, Wedderburn polynomials over division rings, generic irreducibles of the Brauer algebras, and Hopf monads on categories.