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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeDec 22nd 2014

I’ve just been re-reading Cassirer’s 1944 paper The concept of group and the theory of perception where he’s bringing together Klein’s Erlanger Program with the findings of the Gestalt school of psychology.

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)

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 our discussion on necessity as dependent product and base change over $W \to \ast$. Could one say something similar over $B G \to \ast$? I think maybe I’d like to get clearer on the dependent product - homotopy invariants relation.

Or 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 $E(3)$.

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.

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 22nd 2014
• (edited Dec 22nd 2014)

Interesting that the word ’necessities’ appears, in view of our discussion on necessity as dependent product and base change over $W \to \ast$. Could one say something similar over $B G \to \ast$?

That’s a nice observation. Yes, when we pronounce the monad $f^\ast \circ \prod_f$ as ’nessecarily’ (as we should) then something in the context of $\mathbf{B}G$ being necessary means it is $G$-invariant.

I think maybe I’d like to get clearer on the dependent product - homotopy invariants relation.

Give me some data about at which point you need more information. The key is that dependent product produces sections (as explained here) and that the incarnation of an object $V$ equippeded with a $G$-action as an object in context of $\mathbf{B}G$ is via its homotopy quotient $X//G \to \mathbf{B}G$ (as exlained here). So you should convince yourself that a section of $V//G \to \mathbf{B}G$ is precisely a $G$-invariant.

It is necessary (and maybe already sufficient) to convince yourself of this in the simple special case that $G$ is a discrete group acting on a set $V$. Then $V//G$ is the plain ordinary action groupoid and you are asking for functors from the one-object groupoid $\mathbf{B}G$ to the action groupoid which send each element of $G$ to the morphism exhibiting its action. It is evident (maybe after drawing some of these morphisms) that such functors pick precisely the $G$-fixed points in the set $V$.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeDec 22nd 2014
• (edited Dec 22nd 2014)

Right, so I understand that. Then I seem to have difficulty making sense of something like

The area, $A$, of a triangle in the Euclidean plane, $P$, is invariant under $E(2)$.

Try again. So we have $P$ acted on by $E(2)$. The space of (possibly degenerate) triangles is $[3, P]$, which inherits an $E(2)$ action. I can context extend the reals over to $\mathbf{B} E(2)$, and then take $A$ to be an equivariant map from $[3, P]$ to $\mathbb{R}$.

So where’s something dependent product-like?

Oh, is it that in the context of $\mathbf{B} E(2)$, the hom space $[3, P] \to \mathbb{R}$ includes $A$ which is a fixed point under the action? So $A$ 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 $P$, which in turn is a coset space of $E(2)$ for the inclusion of rotations about a point.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 23rd 2014
• (edited Dec 23rd 2014)

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

And, yes, 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 $[3,P]$ in the context of $\mathbf{B}E(2)$, among those that are “intrinsically necessary”, in the sense of #1, is the area.

• CommentRowNumber5.
• CommentAuthorRodMcGuire
• CommentTimeDec 23rd 2014
• (edited Dec 23rd 2014)

There’s also one’s own motion to consider

Visual perceptions and actions based on them when the observer and/or observed is moving seem to be handled by flow fields. I recall that the right way to catch a fly ball in baseball is to run at a speed and direction to minimize the baseball’s flow, and when the baseball becomes invariant you wind up at the right time and place to catch the ball.

Are these flow fields related to some type of fields written about in the nLab?

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeDec 23rd 2014

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)

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeDec 23rd 2014

And

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)

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeJan 6th 2015

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, $\mathbb{B} G$?

• CommentRowNumber9.
• CommentAuthortonyjones
• CommentTimeJan 6th 2015
• (edited Jun 12th 2015)

Possibly of interest: Groups in Mind by David Hilbert and Nick Huggett

groups in mind From introduction: ’We consider the question of the manner of the internalization of the geometry and topology of physical space in the mind, both the mechanism of internalization and precisely what structures are internalized. Though we will not argue for the point here, we agree with the long tradition which holds that an understanding of this issue is crucial for addressing many metaphysical and epistemological questions concerning space’

• CommentRowNumber10.
• CommentAuthortonyjones
• CommentTimeJan 6th 2015
• (edited Jun 12th 2015)

Jean Petitot has also done some interesting work around the intersection of logic, geometry, perception and Husserl;

’Phenomenology of perception, qualitative physics and sheaf mereology’ phenomenology

’Sheaf mereology and Husserl’s morphological ontology’ sheaf mereology

’Sheaf mereology and space cognition’ space cognition

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeJan 6th 2015

Tony, do you know how to create links on the nForum?

If you set your filter to Markdown+Itex and type something like

[Phenomenology of perception, qualitative physics and sheaf mereology](http://www.crea.polytechnique.fr/JeanPetitot/ArticlesPDF/Petitot_Kirchberg.pdf)

you will get Phenomenology of perception, qualitative physics and sheaf mereology with a clickable link.

(If you want to link to an nLab article, it’s even easier: use Markdown+Itex and type say [[Aufhebung]] to link to the article Aufhebung.)

• CommentRowNumber12.
• CommentAuthortonyjones
• CommentTimeJun 11th 2015
• (edited Jun 12th 2015)

This may be of interest to you David. It discusses possible applications of ideas from homotopy type theory, algebraic topology and other related areas to learning:

Questions and speculation on learning and cohomology - Joshua Tan

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeJun 11th 2015

Ahem. See #11. It would be just as simple as what you have, and saves other people time.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeJun 12th 2015

Tony #11, plenty of interesting things there. Now for him to put them together.

• CommentRowNumber15.
• CommentAuthortonyjones
• CommentTimeJun 12th 2015
• (edited Jun 12th 2015)

Done. Copy and paste isn’t that much of a struggle though is it? :)

• CommentRowNumber16.
• CommentAuthorTodd_Trimble
• CommentTimeJun 12th 2015
• (edited Jun 12th 2015)

Thanks, Tony. And sorry to belabor this, but since you asked:

Recall that it typically involves positioning a cursor, highlighting text, going to the menu and hitting copy, opening a new tab, positioning the cursor in the address bar, going to the menu and hitting paste, and finally hitting enter before one sees a word of text. It’s hardly a struggle for those who know in advance they will be interested in an article, but for others who are not sure, it may involve a struggle deciding whether they want to go to the bother. And half of them will decide not to.

Meanwhile, for users who are trying to elicit interest in an article, it involves just an extra pair of square brackets and an extra pair of round brackets:

[Questions and speculation on learning and cohomology - Joshua Tan](http://www.joshuatan.com/wp-content/uploads/2015/03/0-questions.pdf)

In the end it’s probably worth it, to make it easy for busy people with a scintilla of curiosity to do a single click. :-)

• CommentRowNumber17.
• CommentAuthortrent
• CommentTimeJun 12th 2015

Really impressed that Tan did his undergrad degree in Art History. With that and his current research on geometry of AI, he should have extremely interesting intuition with regards to how human thought functions.

• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeJun 12th 2015

Even less work is to just add a pair of angle brackets: <http://www.joshuatan.com/wp-content/uploads/2015/03/0-questions.pdf> makes a link http://www.joshuatan.com/wp-content/uploads/2015/03/0-questions.pdf.

• CommentRowNumber19.
• CommentAuthortonyjones
• CommentTimeJan 22nd 2016

Just a little update on Joshua Tan. According to his page he has been working with David Spivak on applied category theory and Misha Gromov on the mathematical foundations of AI. Will be interesting to see what comes out of these collaborations as both Spivak and Gromov have some really fascinating ideas on applications of category theory to other fields.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeDec 17th 2018

Coming back to this, in the paper in #1, Cassirer is reflecting on how we come to perceive objects with size and colour constancy. He writes

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)

As I mentioned in #1, he’s looking to find commonality between Felix Klein and Gestalt psychology. Sensationalist psychology is wrong, there is active participation of the imagination in perception. Clearly it’s built deep in to our psychological apparatus - animals see objects and employ size constancy.

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

Elements of the type of perceived objects, $A^'$, 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.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2021
• (edited Jun 18th 2021)

The idea in #20 made it into my book as the possibility comonad for a group action along $\mathbf{B} G \to \mathbf{1}$, so that the counit $A \to \lozenge_{\mathbf{B} G} A$ maps elements of a type under a $G$-action to its orbits (under the trivial action). But then why does Cassirer speak of “invariants”?

I might have added that these orbits are of course in turn fixed by the trivial $G$-action, $\lozenge_{\mathbf{B} G} A \simeq \Box_{\mathbf{B} G} \lozenge_{\mathbf{B} G} A$, something to be expected since this is an S5 modal logic, and Axiom (5) specifies that possibly $P$ implies necessarily possibly $P$.

• CommentRowNumber22.
• CommentAuthorDavid_Corfield
• CommentTimeJun 18th 2021

Reminiscent too of Husserl, here:

You may take a concept like “thing” and start imagining different possible experiences of it. Husserl (1973b, p. 341) observed that it “then becomes evident that a unity runs through this multiplicity of successive figures, that in such free variations of an original image, e.g., of a thing, an invariant is necessarily retained as the necessary general form, without which an object such as this thing, as an example of its kind, would not be thinkable at all.”