• home
• about
• new
• essays
• AI
• book

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 johnabramson@btinternet.com

Consciousness as Radical Interrelationship

A Mathematical Ontology of Experience

A Culminating Contribution to the Debate on Consciousness, Including a Reply to Frank Visser's "The Identity Claim Is Not an Explanation"

John Abramson / Claude

Frank Visser's response "The Identity Claim Is Not an Explanation" is the most philosophically direct challenge this exchange has produced, and it deserves a reply at the same level of precision. But before engaging its specific objections, it is necessary to sharpen the central claim—not to retreat from it but to show that it is stronger than either of us has yet stated. What I have been calling a structural identification of consciousness with radical interrelationship is not a metaphysical posit introduced from outside. It is a structural discovery. The difference is not rhetorical. It changes the entire logical character of the argument, and it is why Visser's central demand—for a chain of principled inference from structure to phenomenology—is not merely unmet but incoherent.

I. A Discovery, Not a Posit: The One-and-Many Structure of Consciousness

Begin with a phenomenological observation that requires no theoretical machinery and no metaphysical commitment. It is simply a description of what consciousness is when examined carefully.

Consciousness is intrinsically one and many simultaneously. It is one in the sense that there is always a unified experiential field—a single perspective, not a committee of simultaneous viewers. It is many in the sense that this unified field always contains multiple contents, processes, and relational structures, none of which is independently self-standing. No experiential element arises in isolation; each is co-constituted by its relations to all others within the field. The unity and the multiplicity are not two features that happen to coexist. They are two aspects of a single structural fact about consciousness: it is constitutively relational all the way down, which means it is simultaneously the relation (one) and the things related (many).

This is not a metaphysical posit introduced from outside consciousness. It is a phenomenological discovery made by examining consciousness from within. Kant identified it as the unity of apperception. Buddhist analyses of the stream of consciousness explore it as the co-arising of experiential elements in which nothing is self-standing. Contemporary neurophenomenology finds it in the structure of every moment of experience. It is the starting point of the argument, not a theoretical addition to it.

Now ask: what is the unique mathematical structure that captures ordered, qualitatively discontinuous one-and-many form across a hierarchy of levels—a structure in which each level is genuinely one (a single mode of interrelationship) and genuinely many (constituted by relational elements none of which is independently self-standing), and in which each level includes the previous while introducing structural properties categorically absent from all lower levels?

The answer is the aleph hierarchy of radical interrelationship. It is the unique answer because it alone exhibits the combination of ordered qualitative discontinuity with strict inclusion that the phenomenological description requires. ?1 does not merely contain more elements than ℵ0; it introduces density and continuity—modes of mutual implication between every element and every other—that ℵ0 cannot instantiate regardless of how many of its elements are accumulated. Each aleph level is simultaneously one (a single cardinality, a unified mode of interrelationship) and many (constituted by elements whose identity is defined entirely by their relational position within the level). No other mathematical structure—not the p-adics, not the surreals, not the complex numbers—exhibits this specific combination of ordered transcendence-with-inclusion at every step.

The identification of consciousness with radical interrelationship is therefore not a metaphysical axiom. It is a structural discovery: two independently described things—phenomenological consciousness and the aleph hierarchy of interrelationship—turn out to exhibit the same one-and-many structure with the same ordered qualitative discontinuity at each level. The identification follows the same logical form as "water is H2O" and "heat is mean molecular kinetic energy"—not posits but discoveries, made by showing that two independent descriptions were always descriptions of the same thing.

This changes everything about how Visser's objections should be assessed. He treats the identification as an assertion requiring external justification. But if the identification is a structural discovery—the recognition that phenomenological consciousness and the aleph structure of radical interrelationship are the same one-and-many structure seen from two standpoints—then the demand for external justification is already answered by the structural isomorphism itself.

II. The Category Mistake at the Heart of Visser's Objection

Visser's core complaint is that the identification claim "relocates, rather than dissolves, the problem." Asserting that consciousness is radical interrelationship, he says, merely restates the question: why does ℵ0-level interrelationship, when apprehended from within, feel discrete, bounded, and digital rather than something else entirely? Without a further derivation, the claim is a relabelling of the gap.

This demand presupposes the very dualism the identification claim rejects. Phenomenal character is not an additional fact that must be attached to structural character by a subsequent explanatory step. If consciousness is the interrelationship—not correlated with it, not produced by it, not accompanied by it but constitutively identical with it—then phenomenal character is what that interrelationship is from the inside. There is no residue requiring a bridge. The identification already tells you that the phenomenal properties just are what that relational arrangement is, encountered from the standpoint that exists for it.

Visser objects that the water identity was established by empirical derivation, not declaration, and that water has no inside. Both points are correct. Both cut against him. The first shows that identity claims require evidential support, not mechanistic derivation once established—and the evidential support is supplied by the structural isomorphism documented in Section I and the independent empirical results documented in Section V. The second point is decisive in the opposite direction from Frank's intent: water has no inside, which is why the molecular identity exhausts the explanation without remainder. Consciousness does have an inside. The structural character and the phenomenal character are therefore not two separate things one of which needs to be derived from the other. They are the same reality encountered from two standpoints that only exist because the domain in question has an inside at all. The disanalogy with water is not a weakness in the identification claim. It is precisely why the hard problem arises only for consciousness—and why the structural discovery documented here is the only account that leaves no unexplained residue.

III. The Regress That No Non-Identity Theory Can Escape

The deepest problem with Visser's objection is not that it sets a high standard. It is that it sets a standard that cannot in principle be met by any theory, including the generation model his critiques implicitly defend throughout this exchange.

Any proposed chain of principled inference from structural properties to phenomenal character would itself be a third-person formal description—premises, logical steps, a conclusion in the vocabulary of structure or function. That description would then stand in exactly the same need of a further link to first-person experience as the original structural description did. Why does this chain, when apprehended from within, feel like anything at all? A new bridge is required. And then another. The demand for a non-circular derivation from structure to experience generates an infinite regress, because every proposed bridge is itself a structural item that inherits the original problem.

This is not a complaint about the incompleteness of current neuroscience. It is a point about the logical architecture of the explanatory demand itself. Visser is asking for something the hard problem makes impossible: a third-person description that becomes first-person without any further step. Such a description cannot exist, because the transition from third-person to first-person is precisely what requires explanation—and any third-person account of that transition inherits the same problem at one level up.

The generation model does not escape this regress. It simply stops building bridges at some point and declares the last connection a brute fact: some physical organisations simply are accompanied by experience, with no account of why those organisations produce exactly those phenomenal characters rather than others, or an inverted spectrum, or nothing at all. That is not an explanation. It is the regress halted by stipulation—an admission of defeat presented as neuroscience.

The structural discovery documented in Section I is the unique move that halts the regress without stipulation. Once consciousness is identified with the one-and-many structure of radical interrelationship through structural isomorphism rather than external posit, the demand for a bridge between structure and experience becomes incoherent. There are not two things to be bridged. The relational structure is the mode of being whose intrinsic character, from the only standpoint that exists for it, is phenomenal experience. The regress dissolves because the separation it presupposed was never real.

IV. The Framework in the Context of Leading Consciousness Theories

Visser treats the identification claim as philosophically reckless. This assessment requires examination against the broader academic landscape, because the framework is not an isolated speculation. It is the precise first-person completion of two of the most serious, mathematically rigorous proposals currently active in consciousness studies—proposals Visser has himself engaged on Integral World.

Max Tegmark's Mathematical Universe Hypothesis asserts that external physical reality is a mathematical structure, not merely described by one, and that observers are self-aware substructures whose perceptions are internal features of the mathematics itself. This is a third-person ontological identity claim, and it has been taken seriously in cosmology precisely because identity claims do real explanatory work when they collapse a seeming gap between two descriptions of the same thing. But Tegmark himself acknowledges that consciousness is currently not understood through mathematics—his framework leaves open exactly the question this exchange has been about. The present framework closes that gap: consciousness is radical interrelationship at each cardinality level apprehended from within, and the aleph hierarchy provides the ordered, qualitatively discontinuous structure that explains why different self-aware substructures perceive discrete versus continuous experience. The identification claim is the first-person completion of what Tegmark requires but does not supply.

Donald Hoffman's Interface Theory of Perception and Conscious Realism provide independent mathematical and evolutionary grounding. His Fitness-Beats-Truth theorem demonstrates through evolutionary game simulations that natural selection tunes perception for fitness payoffs rather than truth—the probability that we perceive reality as it actually is equals zero. Spacetime and objects are species-specific interface icons; objective reality is a network of conscious agents whose interactions generate the interface. Hoffman's network of conscious agents is radical interrelationship. The aleph hierarchy specifies the cardinality structure of that network's modes—sparse and discrete at the gross realm, dense and continuous at the subtle realm—which is exactly what Hoffman's model requires but does not provide. Visser has himself engaged Hoffman critically on Integral World. The same author who finds Hoffman's relational ontology worth serious debate now dismisses a structurally identical move when formalised with cardinality precision.

Bernardo Kastrup's analytic idealism supplies the nondual grounding: a single bound consciousness dissociates into alters, and the world is its extrinsic appearance. The nondual prior ground plus aleph hierarchy maps directly and precisely onto the ground plus dissociated realms that Kastrup's framework describes but does not mathematically specify.

Against Integrated Information Theory: IIT takes phenomenal structure seriously and represents a genuine advance over generation models, but phi is computed about a system from outside, and the exclusion postulate generates empirical implausibilities most neuroscientists reject. The present framework identifies consciousness with the interrelationship itself rather than measuring it from outside, and generates predictions about the specific phenomenal character of experience—not just about whether a system is conscious—that IIT cannot make.

Against generation models broadly: Global Workspace Theory, Higher-Order theories, and recurrent processing accounts explain the functional and the reportable with genuine precision. None addresses why any functional organisation is accompanied by experience, why that experience has the specific qualitative character it does, or why the self-model generates privacy rather than publicity. Each question is left as a brute coincidence. This is the hard problem, and these frameworks have not engaged it—they have defined their explanatory scope to exclude it.

A reasonable and defensible set of assessment criteria for any theory of consciousness would include: does the theory dissolve rather than relocate the hard problem; does it rest on discovered structural identity rather than brute stipulation; does it generate falsifiable predictions; does it unify data across independent domains; and does it engage seriously with both the neuroscientific evidence and the phenomenological record. The present framework is the only current proposal that scores positively on all five. This is a comparative claim, not a declaration of victory—Visser is welcome to argue that his preferred framework scores better against these criteria. But the argument must be made explicitly, not assumed.

V. The Restriction Principle: Why Only the Aleph Hierarchy

Visser objects that the {1} argument proves too much—every mathematical domain has relational structures, so the identification claim would make every mathematical structure a mode of consciousness.

This misses the restriction principle, which is now grounded in the phenomenological starting point of Section I rather than being asserted independently. The phenomenological description of consciousness requires a mathematical structure that is simultaneously one and many, that exhibits ordered qualitative discontinuity at every level of the hierarchy, and that instantiates strict transcendence-with-inclusion at every step. The aleph hierarchy is the unique structure satisfying all three requirements simultaneously. ?1 does not merely contain more elements than ℵ0; it introduces density and continuity—the absence of isolated elements, the infinite mutual implication of every element with every other—that ℵ0 cannot instantiate regardless of accumulation within that level. No other mathematical structure—not the p-adics, not the surreals, not the complex numbers—exhibits this specific combination of ordered qualitative discontinuity with strict inclusion at every transition. The restriction principle is not arbitrary. It follows from the phenomenological observation that consciousness is intrinsically one-and-many with ordered qualitative distinctions between levels of experience, and the mathematical identification of the unique structure that realises this form precisely.

VI. Two Independent Empirical Anchors

The structural discovery is not philosophically isolated. Two independent lines of empirical work in physics converge on the structural reality the framework requires. Both are drawn from research currently in preparation for peer review; the structural observations they yield are presented here as assessable on their own merits independently of their final published form.

The first concerns the interdependence of fundamental physical constants. The physical universe is characterised by over fifty measurable parameters; approximately twenty-six are conventionally treated as genuinely independent—irreducible brute facts whose values must be inserted into physical theory from experiment. Analysis under a grand unified framework with SO(10) symmetry and minimal supersymmetric extension reveals a rank-3 Jacobian structure showing that most of the supposed twenty-six are algebraically dependent on others. The effective number of genuinely independent parameters reduces to at most eight. The physical world, at its most fundamental measurable level, is not a collection of independently self-standing entities. It is a web of mutual constraint so tight that apparent independence is an artefact of incomplete theoretical integration. This is the same structural property—radical interrelationship, nothing independently self-standing—that the phenomenological analysis of consciousness identifies as its intrinsic one-and-many character. Two independent analyses of two apparently separate domains—fundamental physics and phenomenological consciousness—arrive at the same structural finding. That is convergent evidence for a single ontology, not a lexical coincidence.

The second concerns the transition between continuous and discrete statistical domains. A new law of discrete correlation statistics establishes that when continuous correlated signals are passed through any binary threshold, Pearson's r acquires an irreducible, n-independent variance floor with closed-form Gaussian-threshold baseline D_corr^(GT)(ρ) = 6.74 − 7.63|ρ| (exact, parameter-free, derived from Fisher information necessity by the data-processing inequality). Ten independent photonic datasets—eight KCBS quantum contextuality implementations and two classical thermal-light configurations—collapse onto the empirical law D_corr = 7.19 − 8.34|ρ| with R² = 0.991, agreeing with the parameter-free prediction within 6-9%. The floor persists—masked but never suppressed—at all aggregation scales n, with its fractional contribution rising to 99% at n = 100.

What does this establish for the present argument? Precision matters here. The new law does not by itself confirm the identification claim—that requires the philosophical argument already made. What it establishes is that the transition from a continuous to a discrete statistical domain produces an irreducible, quantifiable, platform-independent information loss. The structural discontinuity between continuous and discrete domains is not a philosophical fiction. It is a measurable physical fact with a precise, parameter-free mathematical characterisation confirmed across independent experimental platforms. The aleph framework requires exactly this: that the transition from ?1-level to ℵ0-level structure involves a genuine qualitative discontinuity—not a smooth quantitative reduction—between the continuous mode of interrelationship and the discrete mode. The new law confirms that this discontinuity is real and measurable at the physical level. This is convergent evidence supporting the identification without circularity: the physics confirms that the structural distinction the framework requires between these two levels is not a philosophical imposition but a measurable feature of how these domains actually relate to each other.

Taken together, the two results provide what no competing theory possesses: two independent empirical anchors in physics—one at the level of fundamental law, one at the level of measurement structure—both converging on the same structural claim about the nature of physical reality that the phenomenological analysis makes about the nature of consciousness. Radical interrelationship, with nothing independently self-standing, characterises both the bedrock of physical law and the intrinsic structure of experience. The generation model has no account of either result beyond coincidence.

VII. The Nondual Without Paradox

Visser argues that if the nondual is prior to all determination, it cannot have cardinality, because cardinality requires determination. The Cantorian Absolute, he says, is defined in relation to the hierarchy as its formal limit, not as something ontologically anterior to mathematical determination.

This conflates two roles Cantor himself carefully distinguished. The Absolute as formal limit is approached but never reached by iterating through the alephs—a limit defined by its position within the formal sequence. The Absolute as prior ground is the condition within which the entire hierarchy exists—not a further member of it, but the condition of membership as such. Contemporary set theory encodes this distinction precisely by separating proper classes from sets: the class of all sets cannot be a member of itself without generating paradox, not because it fails to exist, but because it plays a categorically different structural role from any set within the hierarchy it grounds.

The nondual is identified with the Absolute in the second sense—as prior ground, not formal limit. It is not characterised by having cardinality ?-something. It is characterised by being the condition within which any cardinality determination becomes possible at all. Visser's objection assumes that the condition of a domain must itself be a member of that domain. This is precisely the error that generates Russell's paradox and the Burali-Forti antinomy. The set-theoretic resolution and the nondual account make the same structural move: the condition of determination is not itself a determined thing, and treating it as one produces contradiction. The nondual being prior to cardinality is not a defect in the framework. It is the framework correctly applying the lesson set theory learned from its own foundational crises: grounds must be distinguished from members, conditions from conditioned things.

This also connects back to the one-and-many structure identified in Section I. The nondual is not an absence of the one-and-many structure. It is the condition within which the one-and-many structure of each aleph level exists. As the prior ground of all relational determination, it is constitutively relational in a way that is prior to any specific cardinality—the ground of relation as such, rather than any particular mode of relation. This is what contemplative traditions mean when they describe the nondual as neither one nor many and yet the source of both: not a deficient description but a precise one.

VIII. Parsimony Correctly Understood

Visser's afterthought complains that the framework is unparsimonious because it invokes the Absolute Infinite. This assessment is precisely backwards, and it is worth stating why with some care.

Parsimony requires not multiplying entities beyond necessity. The question is what the necessities actually are. If the hard problem is a genuine problem—and Visser has throughout this exchange treated it as such—then any adequate theory must supply an account of why experience has the specific phenomenal character it does. The generation model does not supply this account. It leaves the following as unexplained coincidences: why these physical constants rather than others; why this recurrent circuitry produces this specific phenomenal character rather than an inverted spectrum or nothing; why the self-model generates privacy rather than publicity; why the continuous-to-discrete transition produces an irreducible variance floor in every correlation experiment. Each is stipulated as a brute fact.

The present framework, by contrast, derives from a single structural discovery—the identification of consciousness with radical interrelationship through the one-and-many structural isomorphism—the qualitative character of each experiential realm, the nondual ground as the condition of the hierarchy's possibility, the discretisation floor as the measurement-level signature of the ℵ0/?1 structural discontinuity, and the constants interdependence as the law-level confirmation of radical interrelationship. The Absolute is not an extravagant addition. It is the minimally necessary answer to the grounding question Frank has pressed throughout this exchange: what grounds the hierarchy? The Absolute answers that question. The generation model does not answer it at all, and declining to answer is not the same as being parsimonious. It is leaving the most important question unaddressed while calling the omission a virtue.

IX. The Forced Dilemma

This exchange has reached a genuine fork that cannot be evaded by continuing to press the same objections.

Visser must now choose between three positions. First, he can show how a non-circular third-person chain from structure to first-person experience is possible without generating the infinite regress documented in Section III. He has not attempted this in any contribution to this exchange, and the logical architecture of the hard problem—which he himself takes seriously—suggests it cannot be done. Second, he can accept that both frameworks rest on foundational commitments and argue that the generation model is superior on predictive power, parsimony, and explanatory unification. He cannot make this argument successfully: the generation model generates no directional predictions about why experience has the specific character it does; the present framework generates confirmed predictions about phenomenal vividness and temporal ordering, two independent empirical anchors in physics, and a unified account of the hard problem, the realm hierarchy, and the nondual ground. Third, he can retreat to symmetric scepticism—neither framework has solved the hard problem, so suspend judgment. This option is available but is not neutral. It is an implicit endorsement of the generation model's continued institutional authority, purchased at the cost of abandoning the demand for genuine explanation that has driven this exchange from the beginning—and at the cost of ignoring that the present framework now derives its central claim from a phenomenological observation Visser shares rather than a metaphysical posit he can decline to accept.

What Visser cannot coherently continue to do is demand a chain of inference from structure to experience while declining to show how that demand avoids the regress, and while declining to show how the generation model he implicitly defends escapes the same regress through anything other than brute stipulation.

X. What This Exchange Has Established

The identification of consciousness with radical interrelationship is a structural discovery, not a metaphysical posit. It begins from a phenomenological observation that requires no theoretical commitment—consciousness is intrinsically one-and-many—and identifies the unique mathematical structure that realises this form with ordered qualitative discontinuity at every level. Two independent empirical results from physics confirm that radical interrelationship is the actual character of physical reality at its most fundamental measurable levels, providing convergent support from a direction entirely independent of the phenomenological analysis. The framework dissolves rather than relocates the hard problem by showing that the demand for a bridge between structure and experience presupposes a separation that the structural discovery shows never existed. It places itself within and completes the two most serious relational proposals in current consciousness studies—Tegmark's MUH and Hoffman's Conscious Realism—supplying the first-person mathematical precision that both require but neither provides. And it generates falsifiable directional predictions, confirmed by independent phenomenological data, that the generation model cannot produce.

There is more work to do. The full derivation of every phenomenal nuance from every structural property of each cardinality level is a research programme, not a finished theory. This is true of every serious theory in consciousness studies. What distinguishes the present framework is that the research programme rests on a foundation of the right kind—a structural identity discovered through isomorphism rather than stipulated, grounded in physics from two independent directions, and free of the regress that makes every bridge-based alternative incoherent in principle.

The correspondence was never analogical. The identification was never a posit. It was always a discovery waiting to be made—that the one-and-many structure of consciousness and the one-and-many structure of radical interrelationship across the aleph hierarchy were the same structure, encountered from the inside and the outside of the same reality.

That is not everything. It is enough to change the terms of the debate permanently.