What is immediately striking in this bipartisan account is that it suggests a more fundamental division, a prior epistemic split. For, it would seem, materialism most comfortably aligns itself with empiricism (it is through the investigative observation of the similarities between singular pipes that we can arrive at a universal concept of pipeness); whereas idealism appears most comfortably to align itself with rationalism (it is through the intellectual deduction of the universal category of pipeness that we are able to adduce what is shared amongst idiosyncratically singular entities we can then learn to refer to collectively as pipes). A problem, however, here presents itself. If, for materialism, it is the empirical observation of similarities between singularities which are then abstracted to constitute knowledge of universals, then there is a contradiction: since, to define universals as the similarities between singularities is, in effect, to negate the existence of genuine singularities. Singularities disappear given that, in materialism, every singularity is ex post facto demonstrated to be more or less related to other singularities; to, that is, participate to a greater or lesser degree in a universal. In other words, we are faced with a dialectical reversal whereby materialism retroactively passes over into idealism. Contrariwise, for idealism, if it is the intellectual deduction of universals that permit knowledge of singularities, then genuine universality is annulled insofar as universality is never truly encountered in the singularities which are said to instantiate it since, qua singularities, they are by definition singular and not universal, thereby forcing the conclusion that we are faced with a dialectical reversal whereby idealism has always already passed over into materialism. 

The implications of these reversals are not, however, hereby exhausted. For it is furthermore clear that, for materialism, just as there can be no genuine singularity, there can also be no genuine universality either; symmetrically as, for idealism, just as there can be no genuine universality, neither can there be genuine singularity. Why is this? In order for materialism to establish genuine universality, it would have to be able to exclude the possibility of a singularity being discovered which would not be subsumable under any universal, other than the universal designation of singularity as such; whilst for idealism to establish genuine singularity, it would have to be able to exclude the possibility of the deduction of a universal in which a singularity could be shown to participate. Both, strictly speaking, are impossible: the former posits an ability to examine all possible cases, which by definition is an infinite sum and therefore an interminable task; whilst the latter is, obviously, a straightforward contradiction, given that the designation of singularity is itself a universal, given that all singularities minimally and by definition participate in at least one universal – the universal of singularity.

Now from this it is possible to make the following, crucial, extrapolation: we are able to conceive of a singularity that participates only in its own singularity and in nothing else; what we might, in other words, call a universal-singular. A universal-singular is a singularity in which the instant it coincides with itself, it misses itself in a transmogrifying movement towards universality. Indeed, we can see that such non-coincidence with self is true of the inverse, a universal that participates only in its own universality, a singular-universal: at the precise moment a universal expels from itself anything other than itself – which is to say, anything singular – it disappears as a universal in a paradoxical movement towards singularity. Critically, it is in this movement that we grasp how the universal-singular and the singular-universal are not to be collapsed into either universality or singularity – for that would be to construe identity as stasis and self-coincidence, a construal our foregoing analysis has called into doubt. Rather, the universal-singular and singular-universal correspond to the non-space at the heart of identity as such, the space separating identity from itself.