The Key Objection"Mathematics may be defined as the subject in which we never know
what we are talking about, nor whether what we are saying is true"
Isn't this whole approach marred by an unargued strong assumption? It's being assumed that, in explaining existence, 0 can be treated as a number, or at least as a condition analogous to a number, and that this number [or numerical cancellation etc] is in some sense realised as the world? Irrespective of how this conjectural zero ontology is dressed up, there is something substantive that needs to be explained. If the notion of zero objects or events is intelligible, why isn't it likewise intelligible to ask why there aren't zero properties simpliciter? Why are there (allegedly) merely zero net properties as conjectured?
This objection is very much to the point. Yet its rationale - that there is some more 'minimal' possible alternative state of affairs with which to contrast what exists - raises the question of the intelligibility of nothing whatsoever existing far more seriously than before; and certainly not just in the context of a rhetorical argumentative ploy ("I don't know what you mean by..."). One can apparently state the epistemic possibility of nothing having existed rather than something. But it's unclear how it could make cognitive sense to talk of the epistemic possibility of nothing-or-other having even been the case. For the notion of something-or-other being the case is about as conceptually primitive as one can get. For just what is the (supposedly non-self-refuting) alternative with which one would be contrasting the generic notion of existence - in the sense of something-or-other being the case - that we have at present? The notion doesn't seem to make any sense. It's self-stultifying.
And yet if, as this work-in-progress suggests, the net absence of anything at all is supposed to be a sort of condition, then isn't this condition a sort of minimal existence or state of affairs that is itself in need of explanation (as distinct from mere explication)? Fortunately, this may be too quick an inference. Were it the case that none of one's terms referred, as advocates of a more minimal minimalism and more nihilistic nihilism would paradoxically seem committed to claiming, then it's unclear how any such substantive conclusions at all could be drawn. It seems impossible that one could allude to the possibility of comprehensive reference-failure "from the inside" of one's conceptual scheme on pain of incoherence. This minimal cognitivism may of course be too complacent. Total reference-failure seems quite feasible in others - and thus something like it is not obviously impossible, as distinct from inexpressible, in oneself. One hasn't any particular reason to think of oneself as epistemically privileged or uniquely unique compared to other mortals. By parity of reasoning, this can lead to depressingly sceptical conclusions on one's prospects of referential success. Sadly, intellectual history is littered with cheap definitional victories which have turned to ashes. So at best the notion that something must be the case ( 0 - or so I claim) enjoys a paradoxically provisional status in an epistemic if not metaphysical sense.
One may still worry that too rich a notion of 0 is being used. Yet unless one can coherently construct a more minimal alternative, then it may prove the best and most rigorous there is.