I have been ambushed by a cold, consequently scarcely able to think of things more productive than this:
This and this.
The concept of singularity is curiously self-contradictory. It denotes a quality of utter uniqueness. If two objects ‘have’ this quality, then what is unique about them cannot be this – i.e., having singularity. Or in other words: Object A is singular; object B is singular. Do they therefore share some quality, X? if so, then this cannot be what is singular about them.
And yet, how to designate the singular quality without using some name which renders it exchangeable? Indeed, it, whatever it is, would seem in its nature to refuse the span of a name, so that to designate it (the singular quality) is a kind of performative contradiction.
What is singular would seem in some way to be non-linguistic, and to shine forth only when, using language, we have created a kind of circumference of error around the thing in question.
Update. A reader has written to me, relating my reflections to Badiou. I am not entirely qualified to respond fully to his comments, esp. as I'm rather ill, but I reproduce them here for those interested:
re: your comments on singularity, I'd like to draw your attention to the mathematics of the contradiction you describe. It's a variation on Russell's Paradox: "The set of all sets that are not members of themselves." http://en.wikipedia.org/wiki/Russell%27s_paradox There have been several set theoretic attempts to avoid this contradiction, the most important of which is the Axiom of Separation. "Given any set A, there is a set B such that, given any set C, C is a member of B if and only if C is a member of A and P holds for C." http://en.wikipedia.org/wiki/Axiom_of_separation This means that the operation P (in this case looking for, or naming, singularity) is performed only on the members of pre-existent set A, not on the set B (all singular objects) that is the result of this operation. There is an exclusion: one looks for singularity regarding qualities x, y, z, but not for singularity regarding singularity itself. Badiou has an interesting discussion of this in his Theoretical Writings (p. 178-180). He remarks on Russell's paradox that "a certain kind of confidence in the concept is thereby undermined", and concludes from the Axiom of Separation: "it means that existence always precedes the separating activity of the concept."