monad

One way to think of a monad is as a notion of computation over some class of algebras. In programming languages such as Haskell and Opal, the notion of monad is used explicitly to introduce imperative effects into a pure functional regime without compromising important properties such as referential transparency. This is essentially done by reflecting the notion of computation to the object-level of the language itself, rather than relying an evaluation policy (such as call-by-value) at the meta-level to order effects.

This notion of monad has nothing to do with Leibniz monads.

This is not to be confused with the concept of a monad as used in functional programming, where it is usually a way to ensure single-threadedness of data, allowing sequencing while preserving referential transparency (if that's your bag).

In Leibniz' philosophy, monads are the basic units that the universe is composed of. According to him, the monads have been arranged by a perfect God in the best and only possible way. Thus, all suffering and pain lead to better circumstances for everybody. This theory was disputed by the French philosopher Voltaire in his novel Candide, a comedic story which mocks this optimism as not only irrational but also inapplicable.

It's hard for us to imagine a set of entirely independent atomic units that are capable of looking around and indeed feeling hunger, but are nevertheless entirely separate and independent of each other in their own development.

This is hard to swallow.

Nevertheless what was harder to swallow, logically, at the time was the notion of inductive reasoning, which was shattered by Hume and his empiricism.

We should try to see Leibniz's attempt at a new world view in this light, his ability to perceive the essential nature of things (which Kant declares impossible) can be thought of as a sincere attempt to address the issues of idealism vs empiricism, and for this reason alone perhaps he should be granted respite from the sarcasm of Voltaire.

By removing the relational aspect of the universe, Leibniz was showing that the universe could still be modelled cogently, and while his model may seem odd to us (hammer and mirror example above) it nevertheless is considerably more difficult to defeat using Hume's logic, as simply doesn't allow for the same holds.

(thanks to Cletus for the insight)

To expand on x3n0n's writeup above, a monad basically allows us to separate expressions into two categories, those that have effects and those that do not. To those not familiar with the functional programming style, an effect-free statement is basically one that does not change the state of the machine. All it does when evaluated is calculate some value. An effect is often thought of something that does change the state or alter the program flow in some way. This could include raising an exception, altering some state variable, or writing to memory or disk.

Monads are useful since they allow us to analyze a program even if it has effects in it. Without monads, it is relatively complicated to figure out if an expression may raise an exception or not, or to figure out how the state of the machine might be changed by an expression. Monads are popular in Haskell, a lazy functional programming language. (by lazy I mean it uses lazy evaluation).

Mon"ad (?), n. [L. monas, -adis, a unit, Gr. , , fr. alone.]

1.

An ultimate atom, or simple, unextended point; something ultimate and indivisible.

2. Philos. of Leibnitz

The elementary and indestructible units which were conceived of as endowed with the power to produce all the changes they undergo, and thus determine all physical and spiritual phenomena.

3. Zool.

One of the smallest flangellate Infusoria; esp., the species of the genus Monas, and allied genera.

4. Biol.

A simple, minute organism; a primary cell, germ, or plastid.

5. Chem.

An atom or radical whose valence is one, or which can combine with, be replaced by, or exchanged for, one atom of hydrogen.

Monad deme Biol., in tectology, a unit of the first order of individuality.

© Webster 1913.