On page 70 he speaks of the animal chain or series, from the monad to man, ascending from the most simple to the most complex.
Pythagoras held that the unit or monad is the principle and end of all.
The second follows because the existence of one monad involves the existence of many.
In what then do these unities differ from the Uniqueness (or monad)?
It thus is clear that merely stating that matter is passivity in the monad is not the ultimate way of stating its nature.
By wanting to be, the monad makes itself the elephant, the eagle, or the man.
But one monad, as higher in the stage of development than another, makes an ideal demand upon that one.
There is nothing of caprice, of peculiarity, in the content of the monad.
We have previously considered the element of passivity or receptivity as relating only to the monad which manifests it.
In every other monad, the entelechy, or energy, is but one factor.
"unity, arithmetical unit," 1610s, from Late Latin monas (genitive monadis), from Greek monas "unit," from monos "alone" (see mono-). In Leibnitz's philosophy, "an ultimate unit of being" (1748). Related: Monadic.
monad mo·nad (mō'nād')
n.
An atom or a radical with a valence of 1.
A single-celled microorganism, especially a protozoan of the genus Monas.
Any of the four chromatids of a tetrad that, after the first and second meiotic divisions, separate to become the chromosomal material in each of the four daughter cells.
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) ))
(2000-03-09)