NOTE-1163: Isaac — Phenomenal structure: what is it and what is it for?

Contribution

It tries to say what a structuralist science of consciousness is, at a grain fine enough to avoid the platitude that every science is structural. It names three kinds of mathematical structure in experience (metric quality spaces, groups of transformations under which constancies are invariant, vector fields of force), gives each a function (meaning, objectivity, agency), and gives the programme a method (iterative de-idealization of isomorphisms among experience, world and brain) and an epistemology (convergence of independently derived models). Its new element, against Lyre’s similarity structuralism, is that geometry is not enough: the higher layers carry what is distinctive of experience.

Key insight

“I do not represent the category chair as a set of possible chairs, but as a group of transformations across which chair-ness remains invariant” (§2.2). Constancy is invariance under transformations that act within experience, not only in the world, and the gap between the varying “subjective” level (the shadowed part of the wall looks darker) and the invariant “objective” level (the wall looks uniformly off-white), both present at once, is where the subject–object distinction comes from. Following Cassirer, perception “is a process of objectification, the characteristic nature and tendency of which finds expression in the formation of invariants” (§3, quoting Cassirer 1944). On this view the checker-shadow “illusion” is veridical recovery of surface reflectance, not error.

Assumptions

• Structuralism as a stance. “structuralism as developed here is not a theoretical assertation [sic] or result, but rather a stance” (§1). It is judged by the trajectory of its models, not by any one isomorphism.
• Mathematics by conceptual analysis. That constancy has group-theoretic structure, and felt pushes and pulls vector-field structure, is “conceptual” and “not empirical”: group theory “just is the abstract mathematical formalization of the qualitative concept of invariance across transformations” (§2.2), and a vector is “just the mathematical object characterized by magnitude and direction” (§2.3). The empirical questions are how far experience is actually invariant, and whether there are such forces in experience.
• Isomorphism is used loosely, as “structure preserving map”, homomorphisms included. Technical worries such as Newman’s objection are set aside because the term is “suggestive, metaphorical” at the methodological level (§4).
• Indirect perception. It takes the gestalt and Cassirer side against Gibson and Noë: the world as experienced is distinct from the physical environment, and only that distinction makes the search for isomorphisms between them coherent (§3).
• Phenomenal claims by introspection. Its claims about levels of colour experience are offered for the reader to “verify through introspection and reflection”, with no position on the mechanisms of perception (footnote 6).
• Specific character only. It addresses phenomenal experience, “one facet of consciousness” (§1). It says nothing about what makes a state conscious rather than not.

Key results

Nothing is proved or tested. The paper’s claims, in order:

• Geometric structure (§2.1). Quality spaces are bounded regions with topological and metrical structure, measured by just-noticeable differences, ratio and magnitude judgements, matching, bisection and semantic descriptors (Dravnieks’ 146-dimensional odour atlas). Visual space is formally one more quality space. Three dimensions are an idealization for colour: asymmetric matching needs more than three (Niederée 2010), and surface features such as self-luminosity motivate up to five (Evans 1974). Olfaction may need from 6 to thousands.
• Group-theoretic structure (§2.2). Constancies (colour, size, shape, timbre) are invariants under transformations of perspective and context, modelled after Klein’s Erlangen programme. They must be invariants of transformations internal to experience, for two reasons:
  • experienced shape is not a planar projection, since convergence in depth is experienced as depth, and measured perceived angles do not match those of a linear projection (Erkelens 2015; Wagner et al. 2018);
  • the transformations can be realized with no physical analogue: apparent rotation between two static images (Shepard 1984), and mental-rotation times linear in angle (Shepard and Cooper 1982). Constancy is “at best approximate” (Durgin et al. 2012; Daoust 2021). The illumination model is a “rough, toy model”, and Krantz’s Grassmann structures for metamers are strictly groupoids, not groups (footnote 5).
• Dynamic structure (§2.3). Gestalt laws of organization, valence (Lewin’s Aufforderungscharakter) and affordance are forces in experience, modelled by vector fields. Affordances “must have degrees of invitational character which compete, i.e. induce a vector field”, against Gibson’s positive or negative. Quantification is shown only as “proof of concept”: lever-press rates, Warden’s electrified-floor obstruction method, Franchak and Adolph’s frequencies, questionnaires.
• Functions (§3). Contrast within a space makes a quality apt to carry information, and an isolated shade is “not apt to play any representational role”. The similarity structure is antecedent structure co-opted for representation, like the arrangement of pixels or of silver halide. Constancy turns flux into “a stable representation of an objective world”. Gestalt segregation into wholes introduces “baseline meaning in the form of a normative part-whole structure”. Valence fields guide action; the hot pot handle invites the grasp despite the knowledge that it is hot.
• Method (§4). Inner and outer psychophysics (Fechner) become the search for isomorphisms among three domains. Following Köhler and Shepard, these are “functional” or second-order isomorphisms, of relations among representations to relations among their objects (Shepard and Chipman 1970). Worked cases: CIELAB, a homomorphism from spectra into colour experience that fails through hue superimportance (Judd 1970) and context; DKL, an LGN-based space that distorts CIELAB distances and misplaces the unique hues (Schloss et al. 2018; Wuerger et al. 2005). The failures direct de-idealization, on the model of residual-phenomena progress in astronomy (Smith 2014). Loudness is a straightforward phenomenal model whose neural correlate is “mysterious and complex”: firing rate turned out not to be isomorphic to loudness.
• Epistemology (§5). Three answers to how a third-personal science targets experience. Relations between experiences, unlike bare qualities, are made “behaviorally manifest” and so comparable across subjects. Introspection, including phenomenology “front-loaded” into design, keeps the target fixed. And, the author’s “own considered view”, the strongest evidence is model convergence: CIELAB (matching) and OSA-UCS (reflective judgements of equal distance) agree coarsely, with some distortion (Oleari 2001), and models derived from different kinds of behaviour that converge “target an underlying phenomenal experience, because such experience is the least common denominator”. Convergence is “never perfect”, and discrepancies often exceed each method’s own uncertainty, which others take as evidence that no well-defined metric exists (MacLeod and Willen 1995; Decock 2006; Barwich 2019).
• The closing puzzle (§6). The models average over subjects, so their target is a “standard observer”, not any individual’s experience.

Claims

id	claim	strength	support
C1	Phenomenal experience has geometric, group-theoretic and dynamic structure	moderate for geometry, weak for the other two	psychophysical practice for geometry; for the others, conceptual analysis plus cited cases, and the author calls them “mere suggestive sketches” (§6)
C2	The invariants of perceptual constancy are invariants of transformations internal to experience, not of physical transformations or planar projections	moderate	experienced depth and measured perceived angles that match no linear projection (§2.2); apparent motion and mental rotation with no physical rotation (Shepard 1984; Shepard and Cooper 1982)
C3	Group-theoretic constancy structure is what makes experience represent an objective world and induces the distinction between subject and object	weak	Cassirer’s argument and the two levels of colour experience, offered for introspective verification; no criterion for when invariance amounts to objectivity
C4	A quality in isolation cannot represent; representational power belongs to the quality space as a whole	weak	analogy with an alphabet and with pixels; asserted
C5	Affordances must be graded and compete, so they form a vector field	moderate as an argument	”in order to explain action” (§2.3), against Gibson’s binary valence; quantification only as proof of concept
C6	An isomorphism between experience and brain is a regulative ideal of method, not a reductive result	moderate	the CIELAB/DKL and loudness cases, where failures of mapping drive refinement
C7	Convergence of quality-space models derived from different behavioural routes is the strongest evidence that they target phenomenal experience	weak to moderate	the CIELAB/OSA-UCS comparison, olfaction and timbre models; partial convergence, discrepancies acknowledged, no measure of how much convergence is enough
C8	The direct-perception stance robs structuralism of its evidential strength	weak	asserted from the need for two domains to map between (§3)

Concepts

• Quality space: the bounded region of possible phenomenal qualities of one modality, with topological (nearness) and metrical (distance) structure (Clark, Gärdenfors, Rosenthal).
• Group-theoretic structure: invariants of experience under a set of transformations with identity, associative composition and inverses. Here it is “conceptual”, the formalization of invariance under transformation, applied loosely (groupoids included).
• Levels or modes of experience: the “subjective” level that varies with context (aperture mode; the “painter’s attitude”) and the “objective” level that stays constant (surface mode), present together.
• Dynamic structure: forces in experience, as against kinematics, modelled by vector fields over a space: gestalt organization, valence, affordance.
• Behavioural field: Koffka’s experienced space of objects that invite and repel, as against the geographical environment.
• Second-order isomorphism: Shepard and Chipman’s correspondence between relations among representations and relations among their objects; first-order isomorphism, where it exists, is a mechanism for realizing it.
• De-idealization: refining a model of world, experience or brain so that the maps among them improve, driven by residual discrepancies.
• Model convergence: agreement in structure between models of one quality space built from different behavioural evidence.

Connections

The paper looks back on purpose: Koffka (1935), Köhler, Lewin, Cassirer’s “The concept of group and the theory of perception” (1944) and Shepard’s work of the 1970s and 1980s, through Hatfield on visual space and Westheimer (2024) on Cassirer. Among contemporary structuralists it cites Fink, Kob and Lyre (2021) and Lyre (2022) (LIT-1344, read in NOTE-1060) as the convergence it joins, Loorits (2014) on the hard problem, and Decock (2006) for the physicalist reinterpretation of phenomenal spaces. It continues the author’s own papers on psychophysics and the geometry of colour (2013), structural realism for secondary qualities (2014) and “The allegory of isomorphism” (2019). Its affordance field is Reed’s (1993) and Rietveld and Kiverstein’s (2014). Negro (NOTE-1054) and Aldé (NOTE-1055), read in the same batch, attack Lyre’s version and do not cite this paper, and it does not cite them.

Bearing on the record

• Two THEORYs warranted, both Proposed. THEORY-959 states the constancy account (C2–C3) as the paper supports it. THEORY-953 states the convergence epistemology (C7), with the failures of convergence that the paper itself concedes.
• THEORY-880, THEORY-867 and THEORY-868. Isaac is not the similarity structuralist those objections target. He agrees with Lyre that geometry is the first layer, but denies that it is enough (“The geometrical structure of phenomenal experience by itself is not enough to generate meaning”, §3), and never says that an experience is nothing but its place in a space. Aldé’s point that colour-space geometry does not mark the unique hues (THEORY-867) is consistent with Isaac’s DKL case, where a neural space misplaces the unique hues; Isaac treats it as a residual to de-idealize, not as a refutation. His stance is the “methods, not metaphysics” structuralism that Negro (THEORY-868) allows. The comparison is mine.
• QUESTION-255. The question named Isaac’s dynamical structures as candidate material for a formal object that holds unresolved conflicts together. The paper does give one shape for it: a vector field of competing valences and affordances over one experienced space, where the hot handle’s invitation and the remembered heat’s repulsion are experienced together, as forces in one field. But the paper has no cover, no local sections and no gluing condition, and its field composes forces by superposition, which is always globally consistent. So it is a candidate for a kind-(1) reading of felt conflict (one global state that holds both sides), not a model of kind (2). The inference is mine.
• QUESTION-254 and CLAIM-412. The record filed this paper because its group-theoretic layer “induces a distinction between subject and object”, one candidate for what a perspective adds to an interpretation. The reading finds the claim stated, after Cassirer, and supported only by the co-presence of a varying and an invariant level in experience. It gives no criterion that would separate a system that extracts invariants (as any object recognizer does) from one for which there is a subject–object distinction. So it does not answer QUESTION-254. For CLAIM-412, Isaac’s layers are a structural analogue of interpretive organization (contrast, invariance, valence that gives significance and guides action), and like Lyre he addresses only the character of experience, not why any process is conscious (CLAIM-411’s gap remains).
• CLAIM-138. Isaac sets the Newman objection aside, because the isomorphisms are “suggestive, metaphorical” at the level of method. His relations are in practice fixed by named behavioural procedures (matching, equal-distance judgements, descriptor ratings), which is the independently specified relations that CLAIM-138 says concede non-structural input. He can pay that cost because he makes no claim that relations alone fix what an experience is. A further instance for the claim’s parallel; the comparison is mine.
• CLAIM-368. Isaac’s valence field and the record’s frustration of maintained demands are both about competing pulls. The paper’s tea towel, placed over the handle to remove its affordance from the visual field, is a case of managing a demand by altering the experienced field, not by satisfying it. A note for the consolidator, not a correction.
• No instruction for machine-learning practice; nothing for the anthology.

Limitations

• Programmatic. Nothing is tested, and the author says the group-theoretic and dynamic layers “are mere suggestive sketches” and asks “How exactly should they be characterized?” (§6).
• “Conceptual, not empirical” does too much. That invariance can be formalized by groups is analytic; that perceptual constancy has the closure, identity and inverse properties of a group is not, and the paper’s own cases fall short: the illumination toy model’s “inverse” of adding a light is defined by filtering the white baseline, so it works only from that baseline, and Krantz’s structures are groupoids (footnote 5). “Group-theoretic” is used for invariance in general.
• The objectivity claim has no test. Nothing in the paper would show that a constancy-extracting process lacks a subject–object distinction, so C3 cannot fail on the paper’s evidence.
• Convergence is not shown to beat the rival explanation. Models built from different behaviour could converge because they share the stimulus geometry or a common decision process, not experience. The paper’s “least common denominator” argument does not exclude this, and it concedes that discrepancies often exceed stated uncertainties.
• Isomorphism is loose by design. By setting the mathematical conditions aside, the paper cannot say when a map counts as structure preserving enough, or what a failed isomorphism shows, beyond “refine”.
• Averaging. Its own closing puzzle: the models target a standard observer, so whether the programme reaches any individual’s experience is left open.
• Specific character only. Whether a state is conscious is outside its scope.

Open questions

• What property of a transformation structure, beyond invariance, would make it a subject–object distinction? A criterion that some invariant-extracting systems fail would make C3 testable.
• How much convergence, measured against what null (shared stimulus geometry, shared response process), is evidence that models target experience? A comparison of model convergence with and without a shared stimulus metric would bear on C7.
• Can valence and affordance fields be measured as fields, with direction and magnitude at each point of an experienced space, rather than as single rates? The paper’s citations measure scalar strengths.
• Can de-idealization reach individual experience, or only the standard observer (§6)?