RE: Monads

[Patrik Eklund <peklund@cs.umu.se>]

[Note from moderator: this thread has strayed; although this post is
allowed, comments closer to categories (not Kant's) are preferred.]

Reference to Leibniz is nice, and so is going back even more in history. 
Going forward into modern history leads to problems of who actually caused 
what. Probably because we then tend to mix history and politics.

Anyway, also having googled, I found this about Leibniz:

§. 1. Die Monaden (Das Worte Monade oder Monas) wovon wir allhier reden 
werden / sind nichts anders als einfache Substanzen / woraus die zusammen 
gesetzten Dinge oder composita bestehen. Unter dem Wort / einfach / 
verstehet man dasjenige / welches keine Teile hat.

"sind nichts anders als einfache Substanzen"
"is nothing but simple substances"
They are, but it is not a mathematical statement.

"woraus die zusammen gesetzten Dinge oder composita bestehen"
"using which you put them together or compose(!) them together"
Now he is cooking. Monad compositions are important. Leibniz and Beck 
working together, I like it. This is closer to mathematics.

"verstehet man dasjenige / welches keine Teile hat"
"is to be understood as something which doesn't have subparts"
I am sure there are non-trivial monads which are not composed (in Beck's 
sense) by other non-trivial monads. But more interestingely, composed 
monads are indeed monads, and even worse (from leibniz point of 
view) submonads do exist, like the filter monad being submonad to the 
ultrafilter monad (with the astonishing fact, yes, I know, I am repreating 
myself, that their respective algebras are Scott lattices and compact 
Hausdorff spaces).

So, basically I like Leibniz, even if he was wrong at this point. History 
is not easy. We say "Rome was destroyed" and we frequently say by the 
goths. Saying that leads us to ask "how could it be destroyed". Seldom do 
we hear "how could it stay alive so long".

Best,

Patrik

PS "Monas" seems mostly to be used for a sailing boat, the "Kiel", and 
"the Mona" is Louvre in Paris.