I am trying to understand the syntax of:
Aller guten Dinge sind drei.
I think that aller guten Dinge is the subject, but I am confused because it does not appear to be nominative. What is the reason for this or where did I make a mistake?
Contrary to the intelligent speculations expressed by others, this used to be a common construction some time ago and I remember hearing it once in a while in my youth, when I still had contact with old fashioned German. To be precise, the construction is:
Genitive of a set + number = number of members of the set.
Consider this part of a poem by Wilhelm Busch:
Seht, da ist die Witwe Bolte,
Die das auch nicht gerne wollte.
Ihrer Hühner waren drei
Und ein stolzer Hahn dabei. –
Max und Moritz dachten nun:
Was ist hier jetzt wohl zu tun? –
-Ganz geschwinde, eins, zwei, drei,
Schneiden sie sich Brot entzwei
Above, the set was ihre Hühner, her chicken. Since other native speakers seem to have completely forgotten this, here’s one more from grandma’s treasure chest:
Ihrer sechzig hat die Stunde,
über tausend hat der Tag.
Söhnchen, werde Dir die Kunde,
was man alles leisten mag!
So, ihrer sechzig means 60 of them and is very similar to English! This poem, by the way, confirms Goethe was a workaholic.
Through this pattern is so strange for the age group here, that it didn’t recognise it, actually it is much alike contemporary German. Consider this:
Der Mächtigsten drei
might sound unfamiliar to you, but
drei der Mächtigsten
is common. So little has been lost, except for some flexibility in placing the genitive! Also, @Celtschk kindly reminds me, that the pattern allows other subset identifications beyond numbers and provides an excellent example from Goethe:
Der Worte sind genug gewechselt,
Laßt mich auch endlich Taten sehn!
Indes ihr Komplimente drechselt,
Kann etwas Nützliches geschehn.
It should be clear by now, but:
aller guten Dinge sind drei = of all things good there are three
It was already correctly said, that the subject is missing and es would be a natural place holder.
The sentence indeed does not seem to have a proper subject. It could be reordered to
Es sind aller guten Dinge drei.
Here es would be the formal subject. The part aller guten Dinge is in the genitive case. An English translation would be
There are three of all good things.
The of in of all good things may explain the genitive in the German sentence.
"Aller guten Dinge sind drei" is quite an old expression, and it might well not (fully) conform to the rules of todays German grammar. The idea expressed in this forum post seems like a good approach: It could be an ellipsis of a longer phrase like
Die Zahl aller guten Dinge ist drei.
which translates to something like
The number of all good things is three.
But that would leave the question how "sind drei" became plural.
And for good measure, the origin of this expression probably goes back to old Germanic law. Three times a year a thing was held, a governing assembly and court of justice. If a defendant did not appear before this court three times, he could be convicted in absentia. Over the centuries, "Thing" changed to the way more general "Ding" (english "thing"), and today "aller guten Dinge sind drei" basically means something like "all good things come in threes" or "third time's the charm".
»Wessen sind aller Dinge?« stands behind this. Whom are those three things? To whom do they belong? Germans use especially this ›Who’s‹ with many cases, like: »Wem sein Füller ist das?« Answer, grammatically wrong, in Northrhine-Westphalia (Cologne, Bonn): »Mir!«. Or, worse: »Der ist mich!«, instead of »meiner!« or »von mir«. As ›Wessen‹ sounds oldfashioned, you’d say straightforward: »Wem gehört dieser Füller (fountain pen)?«