Mathematics as a Theory of Possible Instances
In contemporary physics, it has become increasingly common to encounter theories that are mathematically rich, internally coherent, and generative of vast spaces of possibility—yet strikingly thin when it comes to phenomenal instantiation. These theories do not merely await confirmation; they often proceed as if confirmation were, in principle, optional.
This post does not argue that such theories are wrong, misguided, or illegitimate. Nor does it rehearse familiar polemics about speculation, falsifiability, or the alleged excesses of modern theoretical physics. Instead, it asks a quieter and more precise question:
What kind of thing is a theory that generates possible instances without generating phenomena?
To answer this, we need a disciplined ontology—one capable of distinguishing, without drama, between mathematical possibility, theoretical instantiation, and phenomenal actuality.
Systems, Instances, and Actualisation
Within a relational ontology, a system is not a collection of things but a structured space of potential—a theory of how instances could be. Importantly, this system is not temporal. It does not precede its instances in time, nor does it await their arrival. Rather, it is a theory of the instance: a specification of what would count as an instance if a perspectival cut were made.
An instance, in this sense, is not a thing that pops into existence. It is a perspectival actualisation: the event of a system being construed as instantiated. Actualisation is thus not a process that unfolds within time but a shift in perspective—from structured potential to event.
Finally, a phenomenon is not a bare occurrence in the world but a construed experience: first-order meaning. There is no phenomenon independent of construal, and no actuality that bypasses this cut.
These distinctions matter because they allow us to say something that is otherwise difficult to articulate:
A theory may be perfectly well-formed as a system of possible instances while remaining entirely empty of phenomena.
Mathematics and the Proliferation of Possibility
Mathematics is exceptionally good at generating systems. Given a small set of axioms and constraints, it can explore spaces of possibility far beyond the reach of intuition, instrumentation, or experiment. In doing so, it produces not predictions but potential instances: configurations that would count as instances if the system were actualised phenomenally.
Nothing in this is problematic. On the contrary, it is one of mathematics’ great strengths. Trouble arises only when this generativity is silently reinterpreted as ontological warrant—when the existence of a possible instance within a theory is treated as evidence that something corresponding must exist, or must exist somewhere, or must exist in principle.
From a relational perspective, this move is not an error so much as a category slip. It confuses:
the internal coherence of a system,
the availability of possible instances within that system, and
the phenomenal actualisation of those instances as events.
Mathematics guarantees the first. It does not, by itself, guarantee the third.
Theorising Without Phenomena
Many influential constructs in contemporary physics now live almost entirely at the level of system and possible instance. They are responses not to recalcitrant phenomena but to tensions within theory-space itself: incompatibilities between formalisms, failures of unification, or desires for mathematical elegance.
In such cases, what is offered is a theory of what could be instantiated, not an account of what has been instantiated. The absence of phenomena is not denied; it is deferred—sometimes indefinitely.
This posture is often defended by appeal to historical precedent: today’s speculative mathematics may become tomorrow’s empirical triumph. That may be so. But the relational point is more modest and more exacting:
Until a phenomenal cut occurs, we are dealing with structured potential, not actuality.
Calling such theories “fictional” would be a mistake. They are neither imaginary nor arbitrary. They are rigorous, disciplined, and often extraordinarily sophisticated. What they are not is phenomenally actual.
Ontological Restraint Without Ontological Anxiety
The value of a relational ontology lies in its restraint. It does not rush to inflate theoretical constructs into entities, nor does it seek to deflate them into mere stories. Instead, it keeps its strata aligned.
A theory can be:
mathematically sound,
theoretically fertile,
indispensable for ongoing research,
and still be ontologically uncommitted with respect to phenomena.
Recognising this is not a failure of realism but a discipline of it. It preserves the distinction between what a theory makes possible and what has been actualised as experience.
In the posts that follow, we will argue that this discipline was once central to physics, was temporarily suspended under very specific conditions, and has since been quietly forgotten. To see how that happened, we must turn to the singular success that reshaped the metaphysical habits of an entire field: quantum mechanics.
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