David Corfield
invariance

Imagine a type AA. You consider it a subtype of BB, such that there’s an action of a group GG on it.

This is *AB\ast \to A \to B and an extension through BGB G.

Noether’s theorem and modality.

Cassirer’s 1944 paper The concept of group and the theory of perception brings together Klein’s Erlanger Program with the findings of the Gestalt school of psychology.

…the search for constancy, the tendency towards certain invariants, constitutes a characteristic feature and immanent function of perception. This function is as much a feature of perception of objective experience as it is a condition of objective knowledge. (p. 21)

If perception is to be compared to an apparatus at all, the latter must be such as to be capable of “grasping intrinsic necessities.” Such intrinsic necessities are encountered everywhere. It is only with reference to such “intrinsic necessity” that the “transformation” to which we subject a given form is well defined, inasmuch as the transformation is not arbitrary and executed at random but proceeds in accordance with some rule that can be formulated in general terms. (p. 26) (“grasping intrinsic necessities” is due to Max Wertheimer)

By their reference to such “good points”, the particular impressions receive a new kind of determination. They lose, so to speak, their “atomicity”, their uniqueness as mere particular items; they unite into groups and totals. (p. 28)

The “images” that we receive from objects, the “impressions” which sensationalism tried to reduce perception to, exhibit no such unity. Each and every one of these images possesses a peculiarity of its own; they are and remain discrete as far as their contents are concerned. But the analysis of perception discloses a formal factor which supersedes this particularity and disparity. Perception unifies and, as it were, concentrates the manifolds of particular images with which we are supplied at every moment…Each invariant of perception is … a scheme toward which the particular sense-experiences are orientated and with reference to which they are interpreted. (p. 32)

He thinks that in mathematics this procedure is taken further, right up to group-theoretic invariance, but that its seeds are there in perception.

Interesting that the word ‘necessities’ appears, in view of necessity as dependent product and base change over W*W \to \ast. One could say something similar over BG*B G \to \ast.

To start with perception, how do we ‘know’ that we’re dealing with the same shape in A of this image? We seem to be able to relate 2D projections of a 3D figure subjected to the actions of E(3)E(3).

There’s also one’s own motion to consider. True to his neo-Kantian roots Cassirer quotes Bühler

The concept of factors of constancy in the face of variation of both external and internal conditions of perception is the realization, in modern form, of that which in principle…was known to Kant, the analyst, and which he stated in terms of mediating schemata.

Perceptual images of objects are products of the imagination. Productive imagination is necessary for objective determination.

(Type: p. 22 quoting Kant, “The concept ‘dog’, for instance, signifies a rule according to which my imagination can delineate the figure of a four-footed animal in a general manner, without limitation to any single determinate figure such as experience, or any possible image that I can represent in concreto, actually presents.” Critique of pure reason.)

HoTT treatment

I wonder what of all this can be given the HoTT treatment.

Given a type of ‘discrete’ impressions, A:TypeA: Type, how do we represent unified perceptions? For an arrow A:1TypeA: \mathbf{1} \to Type, then sometimes we perceive it via a choice of group and so a factorisation, 1BGType\mathbf{1} \to \mathbf{B} G \to Type, through say A A^' with a GG-action, so that the unified perceptions are orbits, elements of the action groupoid, *:BGA \sum_{\ast: \mathbf{B} G} A^'.

Elements of the type of perceived objects, A A^', can’t be given by a single impression. With a free action, an orbit is equivalent to a point, hence the unity of the object.

Robust variation. A rotating image, flying bird.

The area, AA, of a triangle in the Euclidean plane, PP, is invariant under E(2)E(2). We have PP acted on by E(2)E(2). The space of (possibly degenerate) triangles is [3,P][3, P], which inherits an E(2)E(2) action. I can context extend the reals over to BE(2)\mathbf{B} E(2), and then take AA to be an equivariant map from [3,P][3, P] to \mathbb{R}.

In the context of BE(2)\mathbf{B} E(2), the hom space [3,P][3, P] \to \mathbb{R} includes AA which is a fixed point under the action? So AA is ‘necessary’ in the context.

That’s rather like the covariance story of fields (co-shapes) defined on a space-time, but I guess here the domain isn’t subject to its own full automorphism group, but just inherits one from PP, which in turn is a coset space of E(2)E(2) for the inclusion of rotations about a point.

The space [[3,P],][[3,P],\mathbb{R}] inherits an E(2)E(2)-action. The area function is a point in [[3,P],][[3,P],\mathbb{R}] and indeed an invariant point.

It’s much like in the covariance story, or rather like the “co-shape” invariants that we discussed recently. A function on a space of configurations is what in physics is called an observable. Here we are looking at the “gauge invariant observables” on the space of triangle configurations.

Maybe one could pronounce it like this: among all observables on [3,P][3,P] in the context of BE(2)\mathbf{B}E(2), among those that are “intrinsically necessary” is the area.

Gibson and Shepard

Some interesting gleanings from Shepard, Roger N., 1984 ‘Ecological constraints on internal representation: resonant kinematics of perceiving, imagining, thinking, and dreaming’, Psychological Review 91:417-47 (pdf)

…it remained for Gibson to adopt the radical hypothesis of what he called the ecological approach to perception (Gibson, 1961, 1979), namely, the hypothesis that under normal conditions, invariants sufficient to specify all significant objects and events in the organism’s environment, including the dispositions and motions of those objects and of the organism itself relative to the continuous ground, can be directly picked up or extracted from the flux of information available in its sensory arrays.

In the case of the modality that most attracted Gibson’s attention—vision—the invariants generally are not simple, first-order psychophysical variables such as direction, brightness, spatial frequency, wavelength, or duration. Rather, the invariants are what J. Gibson (1966) called the higher order features of the ambient optic array. (See J. Gibson, 1950, 1966, 1979; Hay, 1966; Lee, 1974; Sedgwick, 1980.) Examples include (a) the invariant of radial expansion of a portion of the visual field, looming, which specifies the approach of an object from a particular direction, and (b) the projective cross ratios of lower order variables mentioned by J. Gibson (1950, p. 153) and by Johansson, von Hofsten, and Jansson (1980, p. 31) and investigated particularly by Cutting (1982), which specify the structure of a spatial layout regardless of the observer’s station point.

For invariants that are significant for a particular organism or species, Gibson coined the term affordances (J. Gibson, 1977). Thus, the ground’s invariant of level solidity affords walking on for humans, whereas its invariant of friability affords burrowing into for moles and worms. And the same object (e.g., a wool slipper) may primarily afford warmth of foot for a person, gum stimulation for a teething puppy, and nourishment for a larval moth. The invariants of shape so crucial for the person are there in all three cases but are less critical for the dog and wholly irrelevant for the moth. (Shepard 1984, p. 418)

There are good reasons why the automatic operations of the perceptual system should be guided more by general principles of kinematic geometry than by specific principles governing the different probable behaviors of particular objects. Chasles’s theorem constrains the motion of each semirigid part of a body, during each moment of time, to a simple, six-degrees-of-freedom twisting motion, including the limiting cases of pure rotations or translations. By contrast, the more protracted motions of particular objects (a falling leaf, floating stick, diving bird, or pouncing cat) have vastly more degrees of freedom that respond quite differently to many unknowable factors (breezes, currents, memories, or intentions). Moreover, relative to a rapidly moving observer, the spatial transformations of even nonrigid, insubstantial, or transient objects (snakes, bushes, waves, clouds, or wisps of smoke) behave like the transformations of rigid objects (Shepard & Cooper, 1982).

It is not surprising then that the automatic perceptual impletion that is revealed in apparent motion does not attempt either the impossible prediction or the arbitrary selection of one natural motion out of the many appropriate to the particular object. Rather, it simply instantiates the continuing existence of the object by means of the unique, simplest rigid motion that will carry the one view into the other, and it does so in a way that is compatible with a movement either of the observer or of the object observed.(Shepard 1984, p. 426)

Putting the considerations concerning preference for the simplest transformation that preserves rigid structure together with those concerning the conducive conditions for impletion of such a transformation, I have posited a hierarchy of structural invariance (Shepard, 1981b). At the top of the hierarchy are those transformations that preserve rigid structure but that require greater time for their impletion. As the perceptual system is given less time (by decreasing the SOA [stimulus onset asynchrony]), the system will continue to identify the two views and hence to maintain object conservation, but only by accepting weaker criteria for object identity. Shorter paths that short-circuit the helical trajectory will then be traversed, giving rise to increasing degrees of experienced nonrigidity (Farrell & Shepard, 1981). Likewise, if the two alternately presented views are incompatible with a rigid transformation in three-dimensional space, the two views will still be interpreted as a persisting object, but again a nonrigid one. (Shepard 1984, p. 430)

I guess an example to illustrate the final paragraph of #6 is when you hold a pencil loosely in a ring formed by forefinger and thumb. If you move the ring up and down fast enough, the pencil appears to be bendy.

Have we changed the context (in the type theoretic sense) to a more generous group, 𝔹G\mathbb{B} G, e.g., Euclidean to continuous?

Instead of saying that an organism picks up the invariant affordances that are wholly present in the sensory arrays, I propose that as a result of biological evolution and individual learning, the organism is, at any given moment, tuned to resonate to the incoming patterns that correspond to the invariants that are significant for it (Shepard, 1981b). (Shepard 1984, p. 433)

…although J. Gibson (1970) held that perceiving is an entirely different kind of activity from thinking, imagining, dreaming, or hallucinating, I like to caricature perception as externally guided hallucination, and dreaming and hallucination as internally simulated perception. Imagery and some forms of thinking could also be described as internally simulated perceptions, but at more abstract levels of simulation. (Shepard 1984, p. 436)

Foreshadowing the commutative diagram that I much later proposed (Shepard, 1981b, p. 294), Heinrich Hertz succinctly stated that “the consequents of the images must be the images of the consequents” (Hertz, 1894/1956, p. 2). (Shepard 1984, p. 441)

There must also be a geometric theory of event perception, cf. Similarities in Object and Event Segmentation: A Geometric Approach to Event Path Segmentation, Mandy J. Maguire , Jonathan Brumberg , Michelle Ennis & Thomas F. Shipley

Changes in path direction are related to intention in animate events because path extrema, minima and maxima, will occur whenever a force is applied to the motion of an object (Shipley & Maguire, 2008). When the force is internal it will occur when the actor has embarked on a path towards a new goal. These changes in path direction represent local points of unpredictability in the future path of an object and are thus natural points for attention (Zacks, Speer, Swallow, Braver, & Reynolds, 2007).

Just as arm to torso split happens at sudden change of curvature of outline or of total curvature, so sudden curvature in path suggests collision or internal forces/decisions in animate.

Zygmunt Pizlo

  • Symmetry provides a Turing-type test for 3D vision, (pdf)

  • Pizlo, Z. 2008. 3D Shape: Its Unique Place in Visual Perception. MIT Press.

  • Questions about symmetry and visual perception, webpage

  • Making a Machine That Sees Like Us, (OUP)

  • Unifying Physics and Psychophysics on the Basis of Symmetry, (pdf)

David R. Hilbert, Nick Huggett, Groups in Mind, Philosophy of Science 73 (5):765-777 (2006)

Last revised on March 31, 2021 at 10:13:25. See the history of this page for a list of all contributions to it.