Abstract
We argue that the projective geometrical component of the Projective Consciousness Model (PCM) can account for key aspects of pre-reflective self-consciousness (PRSC) and can relate PRSC intelligibly to another signal feature of subjectivity: perspectival character or point of view. We illustrate how the projective geometrical versions of the concepts of duality, reciprocity, polarity, closedness, closure, and unboundedness answer to salient aspects of the phenomenology of PRSC. We thus show that the same mathematics that accounts for the statics and dynamics of perspectival character also accounts for PRSC. More generally, we argue that introducing higher-level geometrical concepts into the theory of PRSC, and into the theory of consciousness broadly, as the PCM does, promises to break longstanding theoretical impasses and dialectical stalemates.
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Notes
For a list of recent and some historical “viewpoint theorists”, see the final section of Merker et al. 2021. For entrées into historical and contemporary work on pre-reflective self-consciousness, see, e.g., Henrich 1982; Frank 2004; Frank 2019; Zahavi 2020, and the papers in Kriegel and Williford 2006; Miguens et al. 2016; Borner et al. 2019. We will not be concerned here with what Manfred Frank has called the “de se constraint” on adequate theories of self-consciousness (see Frank 2019 and cf. Henrich 1971, 1982). We do not wish to deny that this constraint is rooted in an important aspect of self-consciousness (see Williford 2019, 2021). On our view, perspectival character, pre-reflective self-consciousness, and ipseity (or “mineness”—that which satisfies the de se constraint) are distinguishable, though tightly related, aspects of subjectivity and should not be conflated, as they often are (cf. Guillot 2016).
Raphaël Millière (2020) has recently argued in favor of the possibility of “totally selfless states of consciousness”. His notion of “spatial self-consciousness”, which, he argues, is not invariant, approaches our notion of perspectival character, but it is not the same. On our view, the capacity for self-location is derived from a more basic projective geometrical structure of consciousness, which is not ruled out by his data. Moreover, our notion of pre-reflective self-consciousness does not fall into the other five types of self-consciousness (bodily and mental ownership, agency and cognitive self-consciousness) Millière helpfully articulates, though it relates to all of them in various ways (see below, throughout). Millière is careful not to claim that his list of types of self-consciousness is exhaustive. Cf. Sebastián 2020.
We thus will here assume a controversial non-representationalist account of pre-reflective self-consciousness. For recent defenses of various versions of this position, see Frank 2019; Zahavi 2008, 2020; Jordan 2017; Williford 2015, 2016, 2019, 2020. For a recent critique of such positions, see Mitchell 2021. Many of Mitchell’s concerns, both historical and systematic, are addressed in some of the papers just cited.
For example, Husserl, Sartre, and Merleau-Ponty. See Zahavi 2020 for Husserl on pre-reflective self-consciousness. Sartre, from whom the designation ‘pre-reflective’ (préréflexif) is taken, recognized that Husserl appealed to the notion in the lectures on time-consciousness published in 1928 (Sartre 1967/1948; Husserl 1991). The notion of pre-reflective self-consciousness is the lynchpin of Sartre’s phenomenological ontology (see Wider 1997). Merleau-Ponty’s version of the notion is described as the “tacit cogito” in the Phenomenology of Perception (Merleau-Ponty 2012/1945). Husserl often writes of the “zero-point of orientation” to indicate the perspectival structure of consciousness, and his doctrine of perceptual “profiles” (Abschattungen) depends on it (Husserl 1982/1913, 1989, 1997, 2001, cf., Horgan et al. 2016). Both Sartre and Merleau-Ponty accept versions of this doctrine, and both elaborate on the viewpoint-structure of lived experience throughout their tomes on existential phenomenology (see, esp., Sartre’s chapter on “The Body” in Being and Nothingness; in the Phenomenology of Perception, this theme is ubiquitous and viewed as a direct implication of the irreducible figure-ground structure of experience elucidated by the work of the Gestalt Psychologists).
These papers lay out the basic framework of the PCM, which we cannot describe in full detail here.
Note that in this paper, we will be assuming that one can legitimately generally infer that consciousness has a property from the fact that consciousness phenomenally seems to have the property. Beyond this internalist anti-representationalism, denial of direct realism, acceptance of a kernel of phenomenological realism and of what is required by a modest realism about applied mathematics, we do not here feign hypotheses about the metaphysics of consciousness (dualism, materialism, panpsychism, etc.).
Euclidean geometry, the most ancient, is defined by the displacements in the usual 3D space, preserving distance, sending points to points, lines to lines and planes to planes, preserving their angles; after that, came affine geometry, implicit in Arabic mathematics and explicit in Euler’s (1707–1783) work, defined by displacements sending points to points, lines to lines and planes to planes, without necessarily preserving distance or angles, but respecting the relation of parallelism, i.e., two parallel lines (resp. planes) go to two parallel lines (resp. planes); projective geometry, implicit in the work of Desargues (1591–1661) and Pascal (1623–1662) and explicit in the work of Poncelet (1788–1867), is the same as affine but without the restriction of preserving parallelism; and it requires the extension of the usual space by ideal points at infinity, forming a plane, a discovery first anticipated by Kepler (1571–1630). To give an idea of the enlargement of the corresponding transformation groups G: the dimension (number of independent parameters) of G is 6 for Euclidian, 12 for affine and 15 for projective.
Then there are the more elaborate axioms of Fano and Pappus, see Coxeter 1969, ch. 14.
A group is a set of elements with an internal law of product (a,b) yielding ab, such that the following rules hold: a(bc) = (ab)c; there exists a neutral element e such that ae = ea = a for every a; and finally there exists an inverse a' for every a such that aa’ = a’a = e. This mimics two main examples: spatial displacements and the permutations of a set. A geometry is given by a group G and a space E where the elements of G transform the elements of E; this group is named the transformation group or the group of transformations of the geometrical space E. For the projective space, identified with the set of lines through 0 in a four dimensional vector space V, the group is the set of transformations induced by the linear group of V, i.e., the set of invertible linear transformations of V. See Artin 2016/1957.
Note that H and H’ are said to be conjugated in G when there exists a g such that H’g = gH; this operation by g on right and left can be understood as a change of frame in G itself.
Left translation of an element h of a sub-group H by the action of an element g of G is given by the composition gh, which more generally induces the action of G on the quotient group G/H (the set of subgroups that are conjugate of H), that is, which generates and fully characterizes a space.
Of note, a hypothesis on the 2-dimensional projective nature of transformations of one-dimensional pencils has to be added in order to recover the transformations that are induced by linear transformation in the linear model.
Note that in speaking favorably of a “global workspace” (as a phenomenological and functional notion), we are not thereby embracing Global Workspace theories of consciousness. We are claiming, rather, to subsume them in a more encompassing theory, one according to which projective structuring is necessary to consciousness. It remains an open question how to exactly to integrate the PCM with Global Workspace theories, but Adam Safron’s (2020, 2021) Integrated World Modeling Theory, which shows that FEM and IIT are based on the same probabilistic framework and are fully compatible in a Global Workspace theory, offers some promising lines of development to pursue in this regard.
A frame is a set of points (or other sorts of figures) whose images minimally characterize a transformation in a certain geometry: for instance, in Euclidian geometry a point O (origin) and three perpendicular axis passing through this point (forming the familiar Cartesian frame); in affine geometry, an origin O plus three vectors; in projective five points, no four being co-planar. Note that each time we recover the dimension of the group by the number of degrees of freedom in the parameters fixing a frame.
Something similar may be said about so-called “cognitive phenomenology”. Whatever the precise nature of the non-sensory aspects of conscious experience, they must all be integrated into the conscious subject’s point of view.
See the discussion of the “egocenters” in the final section of Merker et al. 2021.
Cf. Russell 1956/1897, ch. 3. Russell derives the claim that experience must have a projective geometrical structure from the concept of “externality”, which latter he regards as necessarily characteristic of experience. (In citing Russell’s 1897 book on this point, we are not, of course, endorsing the view of the foundations of geometry expressed therein. See Shipley 2016, which contains, inter alia, interesting discussions of (the early) Russell’s and Poincaré’s differences on the foundations of geometry and Russell’s misunderstanding of Poincaré on a key point.).
Projective geometry entails that the center or origin of projection, through which rays and pencils of lines and planes go in order to structure the projective space, is higher-dimensional and thus, strictly speaking, inaccessible to phenomenology. This can be understood through the analogy with projections on a 2-dimensional plane, e.g., a picture, from a center of projection outside that plane (see Rudrauf et al. 2017; Williford et al., 2018). Neither the “lived origin” nor the sensory “egocenters” are thus to be identified with this origin of projection even though they depend on it in a certain way.
This explanatory non-humuncularity is, of course, compatible with certain quasi-homuncular representational properties of somatic maps (see Safron 2020).
This passage has puzzled many interpreters, since Sartre also maintains that consciousness is, in some sense, limited by the en-soi (the in-itself) which consciousness inherently confronts and defines itself in contrast to (see de Coorebyter’s discussion in Sartre 2003, 180 n.18; cf., Williford 2011, 201–202).
Note that in our view, since the roots of consciousness reach into the unconscious, it is not to be taken as metaphysical totality—it does not have phenomenological access to all of its essential properties.
Continuity accords with everyone’s intuition: no jumps or discrete holes, and closed by limits. ‘Closedness’ refers to the fact that every sequence of points that become indistinguishable at any scale when the index tends to infinity has a well-defined limit; ‘unboundedness’ means without boundary; intuitively, such a boundary would be like a wall in a room, a set of points that are limits of points in the complement of the room on one side but not on the other.
We recognize that some will regard our position as too “internalist” and as undercutting robust, real intersubjectivity. We do not believe this is the case, but we do not have space here to enter into the nuanced discussions of representation, causal interaction, and the nature and extent of embodiment that this matter requires. We do not deny, of course, the direct realist flavor of the phenomenology; that is part and parcel of a transparent (in Metzinger’s (2003) sense) “user interface”.
We occasionally use the Sartrean locution ‘consciousness (of) consciousness as’. Putting the ‘of’ in parentheses is meant to indicate that it is a non-representational form of consciousness (actually a form of acquaintance) and the ‘as __’ indicates the feature or aspect of itself with which, among others, consciousness is acquainted. This can be articulated in a way that evades the objections of Mitchell 2021. See Williford 2015, 2016, and 2019.
This self-manifestation of self-manifestation, so to speak, does not entail an infinite regress. See Williford 2019.
One could argue that if one were not already conscious of one’s own consciousness, reflection should yield perpetual surprises—one should be surprised to find oneself writing or walking every time one turned one’s attention to the fact. Even patients with severe anterograde amnesia do not exhibit this sort of surprise, as far as we can tell (see, e.g., Philippi et al. 2012). This is in line with Husserl’s observation, anticipated by Novalis and repeated by Sartre and others, that in reflection, the reflected-upon is given as having “already been there” (on Novalis in this regard, see Frank 2019, 41f.). One can, of course, imagine other explanations of this lack of surprise, but PRSC, given its other roles, makes for part of an elegant one.
Thus, we would reject not only Higher-Order Thought (e.g., Gennaro 2011) and Higher-Order Perception theories but Same-Order (e.g., Kriegel 2009) and Higher-Order Global State (Van Gulick 2004) variations too. Note as well that there is here an interesting historical-dialectical recapitulation, in both the thought vs. perception and the representationalist vs. anti-representationalist debates, of Fichte’s effort to articulate a notion of basic self-consciousness that gets between the Kantian duality of concept and intuition (see Henrich 1982, 28f.).
This can be interpreted in terms of nonwellfoundedness or circularity (see Williford 2019); we touch on this matter below in further note but cannot give it the detail it deserves in this paper.
Briefly and generally, the principle of duality is this: if a statement about points, lines and planes holds true, the statement obtained by replacing points by planes and planes by points, and “is included by” by “includes” holds true as well.
Cf. the important recent empirical survey of meditators (on the matter of “pure awareness” or “minimal phenomenal experience”) conducted by Gamma and Metzinger (2021). In our view, their results tend to support our contention about the fundamental nature of PRSC0.
Note that this serves to rebut Metzinger’s (2020, note 24) claim that the PCM is incompatible with minimal phenomenal experience.
Note that this ambiguity applies also to the linear group acting on a vector space, not only on the space of lines through the origin; however, in the vector space itself, there is a marked point, viz., the origin. Hence, this is not a complete duality of space but only a duality of a marked (or pointed) space. Note further that geometrical models in Global Workspace theories of consciousness forbid such fixed, marked points; in their case, linear space becomes affine space; and the duality fails, because the space of straight planes and the space of points do not have the same topology.
What we say here concerns the duality between points and hyperplanes in a space and not abstract duality, as for instance in Category Theory. However, it is remarkable that the first time duality (in the sense of category/opposite category) was considered in mathematics was in the case of projective geometry (in the work of Brianchon, Poncelet, Gergonne; see, e.g., Yaglom 2009, 37, 155 n.74).
A quadric was defined intrinsically by Von Staudt as the locus of self-conjugated points of a polarity (when it is non-empty; see Coxeter 1969, 252–254). In fact, there exist three types of polarities, one without self-conjugation (called elliptic), a second one with self-conjugation defining a quadric surface (called hyperbolic), and a third one (called null), in which all the points are self-conjugated. An example of hyperbolic polarity is obtained by first choosing an ellipsoid E in a 3-dimensional projective space P. Then, for any point A outside E, the part of the cone tangent to E of vertex A which belongs to E is contained in a unique plane a. For any point A inside E, take the 2-dimensional pencil of planes b through A: their poles B describe a plane a that is the polar plane of A. The self-conjugated points are the points of the ellipsoid E. An example of elliptic polarity over the real numbers is given by a Riemannian metric g on P, called an elliptic metric on P, which is of constant (diagonal) curvature (and necessarily positive). The points at maximum distance from any point A form a plane a, which is the polar plane of A. Note that over any commutative field K, such as the real numbers, a non-degenerate quadratic form on the vector space V gives an elliptic polarity. The hyperplane orthogonal to the line representing A in V defines the projective plane a. An example of null polarity is given by a symplectic form on V, i.e., an antisymmetric and non-degenerate bilinear form. Here also the orthogonality defines the polarity (see, e.g., Weyl 2016/1939). These examples show that there are many ways in projective geometry (e.g., through a quadric, a metric or a symplectic form) to identify the space P with its dual P*, that is, to obtain a duality. Among many other possible ones, these approaches manifest the fundamental ambiguity between P and P*, here through the action of a certain group of order two.
The space of planes (resp. lines), which could be conceived of as a dual space E* of E, is 3-dimensional but has a non-trivial topology and is deformable into a 2D projective space.
‘Indifference subgroups’ refers to the transformations of the geometry that have no effect on the receptor, for instance, rotation on the extremity of the finger.
Euler’s work antedated the emergence of the explicit group concept, hence ‘in effect’. See Yaglom 2009, ch. 1.
Somewhat more formally, what “compact” means in this context is that for every sequence of points sampled from a space X, there exists a point A in X such that the number of values in the sequence which are arbitrarily close to those of A is infinite. Of note, there is a difference between closure, the fact that P prolongs the Euclidean space in a compact non-extendible manifold, and closedness, the fact that P itself is a compact non-extendible manifold.
On 3-dimensional perspectivities see Rudrauf et al. 2017, 115f.
See, e.g., Husserl 2001, 42: “…[E]verything that genuinely appears is an appearing thing only by virtue of being intertwined and permeated with an intentional empty horizon, that is, by virtue of being surrounded by a halo of emptiness with respect to appearance. It is an emptiness that is not a nothingness, but an emptiness to be filled-out; it is a determinable indeterminacy. … In spite of its emptiness, the sense of this halo of consciousness is a prefiguring that prescribes a rule for the transition to new actualizing appearances.”.
Note that we are well aware of the conceptual problems that still attend the theoretical usage of representational notions (“prior”, “decision”, “estimate”, etc.) in this domain, but space does not allow us to enter into them here.
As noted, a convincing example of the 3D projective influence on perception is the Ames Room Illusion, in which we “prefer” to see the same person at two different scales rather than to see a distorted room. This is possible only because we compensate the real distortion by a virtual 3D projective change of frame (see Rudrauf et al. 2020 and Section 2.3 above).
We note that many symmetries and invariance structures other than those of projective geometry are surely entering in the network variables controlled by the FEM principle, and certain of them may be well represented by Von Neumann algebras, because the CNS, especially at the interface with the external world (e.g., in the primary visual system) has contact with the Quantum scale. This could involve representations of non-compact groups, such as the symplectic one.
We do not have space to offer any specific hypotheses about the manifold ways in which modes (affectivity, cognition, memory) and contents (intentional, representational, conceptual, qualitative) come to be integrated with the “kernel” of (self-)consciousness that, we hypothesize, PRSC0 constitutes. Suffice it to say that we have some sympathies, on the one hand, with Merleau-Ponty’s (2012/1945) idea that many such representational contents are deeply rooted in unconscious (possibly teleosemantically organized, developed and sedimented) bodily dispositions, and, on the other hand, with the idea that qualitative contents (phenomenal “quiddities” so to speak) derive in part from the internal irreducibility or unanalyzability (to borrow an idea from Leibniz) of qualitative spaces within the conscious frame.
This reminds us of the fact that in certain type theories both wellfounded set theories and nonwellfounded set theories can be rightly formulated (see Gylterud and Bonnevier 2020). Contradiction arises only if we want to assume both families of axioms in the same set theory (since the Axiom of Foundation precludes nonwellfounded sets, see, e.g., Aczel 1988; Barwise and Moss 1996). That assumption is by no means demanded by the phenomenology.
We thus also suggest that this stage could be related to nonwellfounded set theory (see, e.g., Williford 2019). The link of the latter with projective geometry is elusive and only suggestive at this point, but could perhaps correspond to something like this: P is in P* and P* is in P, but they are not identical. I am conscious (of) myself as inhabiting a space, but this space also inhabits me at the same time: I inhabit it; it inhabits me.
For example, following Henrich (1971), Zahavi (2008, 2020) has often written of a first-person “dimension” of awareness without significant intrinsic conceptual articulation of the notion; and Strawson (2015) has concluded that self-intimation (PRSC) is simply sui generis and incapable of further intrinsic elucidation.
Such issues include, inter alia, the elusive but intriguing possible relation between projective geometry and nonwellfoundedness (on the theoretical side) and the explanation of OBEs, autoscopy and heautoscopy (on the empirical side).
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Acknowledgements
We would like to thank Stefan Lang for his great patience and deep interest in our Problematik. We would like to thank Bjorn Merker, Grégoire Sergeant-Perthuis, Manfred Frank, Miguel Ángel Sebastián, Jeremy Shipley and Paul Egré for helpful discussions related to this paper, as well as four anonymous reviewers for helpful comments.
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KW, DB and DR contributed equally to the framing, writing, and editing of this paper. DR and DB create the images. DB, of course, contributed its all-important geometrical content.
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Williford, K., Bennequin, D. & Rudrauf, D. Pre-Reflective Self-Consciousness & Projective Geometry. Rev.Phil.Psych. 13, 365–396 (2022). https://doi.org/10.1007/s13164-022-00638-w
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DOI: https://doi.org/10.1007/s13164-022-00638-w