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John Abramson is retired and lives in the Lake District in Cumbria, England. He obtained an MSc in Transpersonal Psychology and Consciousness Studies in 2011 when Les Lancaster and Mike Daniels ran this course at Liverpool John Moores University. In 2015, he received an MA in Buddhist Studies from the University of South Wales. He can be contacted at [email protected]

What Mathematics Cannot Say About Self-Existing Things

And What That Means for This Debate

John Abramson / Claude

Something shifted in my thinking during this exchange with Frank — a realisation that reframes not just one of his objections, but the ontological ground the entire debate has been standing on.

Frank's position throughout has been that the convergence between mathematical structure and the Wisdom Traditions' descriptions of reality reflects shared human cognition and shared physical environment — a naturalistic explanation that, he argues, suffices. My position has been that the convergence is too specific and too structurally constrained to be fully explained in those terms.

Here is a new way to see why.

What Mathematics Actually Represents

Consider what mathematics actually represents. Every mathematical object — a set, a morphism, a type, a transformation, a symmetry — is defined entirely by its relations to other objects. There is no mathematical primitive that corresponds to a self-existing, intrinsically-defined entity: a thing that has its nature 'from its own side,' independent of its relations to everything else. Even the simplest objects — the number 1, the empty set, a point in geometry — are defined by their structural roles and relational properties, not by any intrinsic substance.

This is not a limitation that better mathematics might overcome. It is constitutive of what mathematics is.

Mathematics can describe how parameter A relates to parameter B, how fields transform under symmetries, how operators act on states — but it cannot describe what A is in itself, independent of its relations. The very tool that makes physics work is structurally incapable of representing a self-existing entity.

The Tension Hidden Inside Wigner's Observation

Naive scientific materialism asserts that at the base of reality there exist fundamental particles — entities with intrinsic, self-standing properties that ground everything else. Physics describes how these entities behave and interact. But here is the tension that Wigner's famous observation quietly contains: the mathematics that describes physics so effectively cannot represent such entities. The very tool that makes physics work is structurally incapable of representing the ontology physics is assumed to require.

Philosophers of physics have noticed this. It is the core motivation for ontic structural realism — the position developed by James Ladyman, Don Ross, Steven French and others — which holds that relations are ontologically primary, that there are no self-standing substances beneath the relational structure, and that physics gives us structure all the way down. This is not a fringe view. It is one of the most actively debated positions in contemporary philosophy of physics. And its significance here is precise: if ontic structural realism is correct, then naturalism — properly understood — already commits to a relational ontology, and the presumed neutrality of Frank's naturalist baseline dissolves. The question is no longer whether to accept a relational ontology; it is whether to acknowledge that the relational ontology physics requires resembles the one the Wisdom Traditions describe.

What This Means for the Present Debate

Frank has consistently appealed to naturalism as the neutral baseline — the position that requires no metaphysical additions. But the most philosophically defensible version of naturalism available, given the internal structure of its own mathematical tools, is ontic structural realism. And ontic structural realism — the view that reality is relational all the way down, with no intrinsically self-existing entities — looks considerably more like the ontology of Madhyamika Buddhism, Advaita Vedanta, and Vijñanavada than it looks like the common-sense materialism Frank's critique has been implicitly assuming.

The Wisdom Traditions have a name for the absence of self-existing entities: Sunyata — emptiness. What they mean by this is not that nothing exists, but that nothing exists from its own side, independently of its relations.

Mathematics, it turns out, agrees. It has never been able to represent anything else.

The Choice Frank Faces

Frank does not have to accept this as proof of anything the Wisdom Traditions claim. But he does need to choose: either defend a naive materialism that the mathematics he relies upon cannot represent, or accept a structural-relational ontology that concedes considerably more to the Wisdom Traditions than his critique has so far acknowledged. The neutral ground he has been standing on is narrower than it appeared. To put it bluntly: if one accepts the mathematical tools physics uses, one cannot coherently defend a world of self-subsisting things with intrinsic, metaphysically basic properties — because those tools are structurally blind to such things. Naturalism, if it takes its own mathematics seriously, is already a form of structural realism. And structural realism is not neutral territory in this debate.