The evolution of topology in the visual field: 2

Part 2

Likewise the experience of movement and colour in the visual field also develop in stages. The primitive visual field may contain only pure motion, usually rotary. Colours appear first as space and film colours. These only later enter their objects. Objects are often seen first as parts (e.g. the handle only of a teacup) that join up together later.

All this complexity may readily be explained on the grounds that we are observing the successive stages of recovery of an image-constructing representative mechanism that has several different basic parts (one for motion, one for colour and one for shape). It is difficult, for me at least, to think of it as representing the gradual recovery of the direct awareness of external objects that naïve realism supposes. How can a film colour be regarded as ‘the way we see” the colour located on the surface of e.g. a rose?

So, this data suggests that, topologically, the basic visual field is a pure formless expanse of nothing. Formal geometrical shapes (lines, curves, triangles, etc) only develop later. So what else can we say about its topology? In my book “Analysis of Perception” I described how fact that the boundary of an ordinary after-image forms a Jordan curve (that separates the whole of introspectable visual space into one ‘inside’ and one ‘outside’) can be used to define a basic topological property of visual space. However, no entire visual field itself (including that of Poppelreuter’s stage 1) forms a Jordan curve, as there is no ‘outside’. So can we say that in imagination a patient could “draw” a visual image of a Jordan curve somewhere on the P. stage1 primitive visual field, as we can on a normal visual field? It is difficult to say. No experiments, as far as I know, have been done to determine what visual imagery is available to patients in Poppelreuter’s stage 1. However, as we know that visual imagery uses the same brain mechanisms as are used in vision, it seems unlikely that such patients posses the required imagery. So can a formless spatial expanse be said to possess any kind of topology? If so, what? More clinical investigations of Poppelreuter’s stage 1 patients might yield some interesting results. It is also possible that the development of vision in very early infancy may follow a similar course but this would be difficult to determine.

There is, however, another topological relation that may be relevant. We can ask if visual space, however primitive and so long as it is ontologically a basic ‘given’ (i.e. a real space), is topologically ‘inside’ or ‘outside’ physical space (as an entirety), or any part of physical space (e.g. brain space). To this question naïve realism, the Identity Theory, and material dualism give different, but definite, answers. So perhaps that gives us a clue as to the basic topology of visual space? Not which answer is true, but that such a question can be asked at all.

The evolution of topology in the visual field: 1

Part 1

One of the major contributions of the great Viennese neurologist Paul Schilder to visual science was the concept of the ‘primitive’ visual field. This concept, which may have profound implications for our topic, was based in part on data from cases of cortical damage, mostly WW1 injuries studied by German neurologists. In such cases it might naively be supposed that, when vision returns, it does so in the manner of first seeing one’s surroundings as we normally see them, but more faintly and fuzzily, with gradual clarification over time. But that is not what happens at all.

As Lord Brain said ”The researches of the last fifty years however have shown that “seeing” in the sense of awareness of sense-data is not an all-or-none process but a progressive integration and discrimination. Poppelreuter’s stages have been described above [here below]. Whether or not this particular series is accepted there is ample evidence in favour of the principle.”

Following severe injury to the visual cortex at first there is no vision at all, not even blackness. Then, in the case of the perception of geometrical form, Poppelreuter’s first stage is experienced as “the visual field, pure and simple, i.e. visual extension without form” [that is a uniform formless field without colour: not even black which is a colour].

Stage 2. The field becomes differentiated so that ‘left’ and ‘right’ (without form) are distinguished.

Stage 3. A surface area becomes further differentiated without distinct dimensions “It appears neither horizontal, nor vertical but the same dimension on all sides”.

Stage 4. The mass develops a direction within the visual field.

Stage 5. Indeterminate forms develop: the mass is perceived as “..somewhat long, small, horizontal, etc.’.

Stage 6. One mass differentiates into several different ones.

Stage 7. ‘Finally, form is perceived in the strict sense, and there is distinction of straight lines, curves, geometrical figures, etc..”

Apparent Distortions in Photography and the Geometry of Visual Space

by

Robert French EMAIL robert.e.french@gmail.com

Abstract:

In this paper I contrast the geometric structure of phenomenal visual space with that of photographic images. I argue that topologically both are two-dimensional and that both involve central projections of scenes being depicted. However, I also argue that the metric structures of the spaces differ inasmuch as two types of “apparent distortions” – marginal distortion in wide-angle photography and close-up distortions- which occur in photography do not occur in the corresponding visual experiences. In particular, I argue that the absence of marginal distortions in vision is evidence for a holistic metric of visual space that is spherical, and that the absence of close-up distortions shows that the local metric structure possesses a variable curvature which is dependent upon the distance away of objects being viewed.

In this paper I will compare and contrast the geometric structure of visual space, which I am using to refer to the space of phenomenal visual experience, and the spatial structure of photographic images. I will argue that topologically both are two-dimensional and that both involve central projections of scenes being depicted. However, I will also argue that the metric structures of the spaces differ inasmuch as two types of “apparent distortions” occurring in photography do not occur in the corresponding visual experience. In particular, in this regard I will take note of both so-called “marginal distortions” in wide-angle photography, and alo so-called “close-up” or “perspective” distortions. I will argue that the absence of these “apparent distortions” in visual space shows that the space possesses a non-Euclidean (in the sense that not all of Euclid’s postulates apply to the space) metric structure.

Before proceeding further it may be helpful to make a few comments on the way in which I will be using the term “apparent distortion.” In the photographic literature¹ a distinction is often made between so-called “real” and so-called “apparent” distortions. “Real distortions” are defined as deviations from a rectilinear perspective due to defective lenses, as for example occur with the “curvilinear “ distortions of wide-angle photography whereby straight lines are projected as being curved near the margins of the resulting photographs; diverging apart in pincushion distortion and converging together in barrel distortion. In contrast, “apparent distortions,” are defined as deviations between the geometric character of visual experiences and the geometric character of corresponding photographs even when no defects in photographic lenses are involved; i. e., where straight lines are projected as being straight in the photographs. I will now turn to some comments comparing the topology of visual space and photographic images; in particular on the issue of the dimensionality of the spaces.

While, due to the existence of visual depth perception, it is often held that visual experience, unlike photographic images, are three-dimensional. I believe that this is clearly not the cast. The standard recursive defintion² of dimensionality is not directly applicable to either case, being an “in the small” criterion in terms of the dimensionality of bounding spaces for all infinitely-small neighborhoods of points of the space in question; the dimensionality of the original space being one greater than that of the bounding spaces. Inasmuch as these neighborhoods will be too small to be visually distinguishable (the limit of phenomenal visual spatial acuity being approximately one minute of arc), it is not possible to strictly apply the criterion to visual space. However, it is possible to apply an analogous “in the large” criterion which was given earlier by Poincaré³ whereby the dimensionality of a space will be one greater than that of the dimensionality of the bounding space for any given region. Since regions of both photographs and also visual experiences can be bounded by a one-dimensional space (a line constituted by the boundary between a projected object and its background)l, it follows that the spaces themselves must be two dimensional.

While counterexamples can be given to Poincaré’s analysis, such as in the cases of a point giving boundaries to two cones at their intersection, or a circle giving boundaries to a solid sphere embedded in a solid torus, it would seem that neither vision nor photography have much in common with these types of cases. In fact it can be noted that in the one case where prima facie it might be thought that visual experience is three-dimensional, when features of objects depicted possess different physical depths from the eye, this very fact can be used to constitute an edge, which constitutes a bounding line around the features in question. Further considerations pointing to the two-dimensionality of both visual space and photographic images include the points that, except in cases of semi-transparency, we neither see the interiors of objects nor intervening points between our eyes and objects being seen, and that these do not appear in photographs either.

A second feature held in common between visual space and photographic images involves the fact that they both correspond to the results of forming a central projection of what is depicted onto a two-dimensional surface. Of course in the visual space case I do not mean that an optical image is literally formed by means of focusing light rays on the space, but rather that the geometric structure of images depicted in the spaced is isomorphic with that which would be formed by such a projection. Thus, in both the case of photography and that of visual space the image is such as would be formed by having a pencil of straight lines (e. g., light rays) pass from each exterior point on the front surfaces of the objects being depicted through a fixed point (the center of projection), and then cutting this pencil of straight lines by a two-dimensional surface at some location either before or after the fixed point. If the cut occurs before the fixed point the projection is known as a direct linear projection, and if it occurs afterwards it is known as an indirect linear projection. It can be noted that in indirect linear projections (which occur on the negative in photography and also on the retina of the eye), the image is in reverse.

Due to the fact that both visual space and photographic images constitute central projections of what is depicted, it follows that both topologic and metric invariants of projection will be held in common between vision and photography. In particular, note can be taken of such topologic invariants as (aside from complications involving occlusion) the total number of objects being depicted and the number of sides possessed by individual objects being depicted. Even while rectangles, when viewed askew, will be projected as trapezoids and circles, when viewed askew, will be projected as ellipses, it can be noted that there are still metric invariants here, notably the “cross ratio.”⁴

Rather than develop in more detail the just-mentioned issue of the nature of invariants of projection in the geometric structure of what is depicted, I wish to instead address issues concerning the shape of the two-dimensional surface which in vision cuts the pencil of rays of a projection. Is it a flat plane like the film of most cameras? Is it a cylinder like the film of a panoramic camera or the screens where wide-angle movies are shown? Or is it a sphere like a planetarium or the retina of the eye? Or is it a still more complex surface? By taking note of two “distortions” which occur in photography but not in vision (and where thus the geometric structure is not invariant between the two), I believe that it can be demonstrated that it is a more complex surface than any of these. I will first take note of a distortion, “marginal distortion,” which takes place in peripheral regions of wide-angle photographs in order to show that holistically the metric structure of visual space is spherical, and will then take note of a second type of distortion, sometimes referred to as “close-up” or “perspective” distortion, in order to show that when objects depicted possess different physical depths from the eye, visual space possesses a variable curvature.

It can be noted that in central projections onto planes, as in most photographs, straight lines are projected as straight lines. However, areas in peripheral regions of a wide-angle photograph are not, as in the corresponding visual experiences, proportional to the visual angles subtended, but instead contain “marginal distortions” whereby areas in peripheral regions of the photograph subtend significantly smaller visual angles than do equal areas near the centers of the photographs. Since the field of view of vision is approximately 180^o by 120^o, which is greater than the field of view of even ultra-wide-angle lenses, I take it that the fact that marginal distortions do not occur in vision is evidence that the metric structure of visual space is fundamentally different from that of a flat photograph. Before proceeding to make some positive remarks on what the metric structure of visual space is, I will enter into a short digression into the geometry of marginal distortion.

To simplify matters concerning the optics of wide-angle lenses here, I will consider the geometric optics of a pinhole camera. It can be noted that the total length of a line lying along the film in such a camera is proportional to the tangent of the angle Θ formed between a line perpendicular to the film passing through the center of projection (the pinhole) and the line lying along the film. Thus, the marginal distortion present at a given angle Θ here will be proportional to the derivative of the tangent of this angle, or the secant squared. Corresponding areas subtended on that region of the film will hence be proportional to the secant of Θ raised to the fourth power. In wide-angle photography the result is analogous to what occurs in polar regions of the Mercator projection of the globe (where a sphere is projected onto a cylinder, which is then flattened out) whereby the polar regions are disproportionately enlarged (e. g., with Greenland being projected as being larger than Australia while in fact possessing less than one third of the area). By similar reasoning, it can also be noted that a sphere in the periphery of a wide-angle photograph will be projected as an ellipse, although this effect does not occur in vision, as L. P. Clerc note as follows:

“From whatever angle we may look at a sphere its outline always appears exactly circular. On the contrary, the plane perspective of a sphere is an ellipse, except in the case where the center of the sphere is on the perpendicular from the viewpoint to the projection plane. As the visual ray to the center of the sphere makes an increasing angle with this perpendicular, so the distortion also becomes greater.⁵

It is true that in extreme wide-angle photography many of the foregoing effects can be avoided with a fish-eye lens, (where the effects are analogous to those present in a conic projection of a globe whereby a projection onto a cone is flattened out so as to form a circular image) but then, due to very-pronounced barrel distortion, straight lines are no longer projected as straight lines. In fact it is noteworthy that in vision also, unlike photography utilizing a rectilinear perspective, straight lines are projected as great circles which converge at both poles. For example, if a long rail fence is looked at head-on then the top and bottom rails will appear to converge towards each other in both directions although this effect will not be present in a wide-angle photograph; it being compensated by the marginal distortion which it was just noted also occurs in these photographs. While it is true that such a convergence also occurs with photographs taken with a fish-eye lens due to the just-noted barrel distortion in the resulting photographs, unlike vision, these photographs will also possess a latitudinal “stretching” in peripheral regions in the sense that the projections of objects along azimuthal (circumferential) axes will be disproportionately large compared with the projections along the polar axes.

It can be noted now that in cases of projections onto spherical surfaces equal solid angles of the projection will subtend equal areas of the sphere. Hence it will both be the case that no marginal distortions will occur in these projections and also that there will be no disproportionate “stretchings” in projected images such as those occurring with the projection of a fish-eye lens onto a flat surface. I take it then that the fact that neither of these “apparent distortions” is present in vision is strong evidence that at least the holistic metric structure of visual experience corresponds to the metric structure of the surface of a sphere.

I wish to turn now to a second type of “apparent distortion, “close-up” or “perspective” distortion, whereby close objects appear to be disproportionately large in a photograph. A good example here is that of a close-up photograph of a human face where the nose appears to be disproportionately large. Even taking into account the comparison between the position of the camera in taking the picture and the distance from which the corresponding photograph is viewed, (the proper viewing distance is determined by the focal length of the camera lens, the distance from the lens to the film when the lens is focused on infinity, multiplied b the extent to which the photographic print is enlarged), this effect is still quite pronounced, as Nelson Goodman notes as follows:

“And even we who are most inured to perspective rendering do not always accept it as faithful representation: the photograph of a man with his feet thrust forward looks distorted, and Pike’s Peak dwindles dismally in a snapshot. As the saying goes, there is nothing like a camera to make a molehill out of a mountain.⁶

Unlike Goodman, I do not believe that the foregoing effect demonstrates any claims that seeing in central perspective is a matter of habit and thus a learned phenomenon that may not have been universally acquired. Instead, I believe that what is actually going on here is related to various “constancies” noted by Gestalt psychologists; in particular, the tendencies towards size and shape constancy. I wish to take particular note here of what Robert Thouless⁷ has termed the “phenomenal regression to the real object,” whereby he claims that perceived size and shape possess intermediate values between those resulting from a projection onto a surface of constant curvature and the real physical sizes and shapes of the objects. Care must be taken in interpreting results of experiments here, and particular note should be taken of the nature of the instructions given to the participants in the experiments. For example, if the instructions are to make estimates of the actual sizes and shapes of the objects much more constancy is reported than in the case of “projective” instructions where the participants are asked to take the attitude of an artist and report the geometric nature of appearances. It is noteworthy that even with projective instructions a significant degree of size and shape constancy is reported.⁸ It is also noteworthy that the reported constancies are significantly greater under binocular vision than under monocular vision.⁹

I will now show how the foregoing results can be accounted for in terms of the internal metric structure of a space possessing a variable curvature and will also show how, at least in principle, phenomenal visual depth perception may also be accountable for by this structure. The key point here is that since the area on a sphere subtending a given spherical angle is a function of the square of the radius of the sphere, the tendency towards sized constancy can be accounted for in terms of the effects of distorting a sphere by means of a “depth function” defined as a function of the physical distance away of perceived objects. Thus, the depth function p can be defined here as

ρ = f(Θ, Ф, t) [1]

Where ρ is the physical distance from the “cyclopean eye” (an imaginary eye centered between the two eyes) to the physical object being seen in the direction Θ, Ф at time t. While in normal circumstance a unique object will be determined by these conditions and thus the depth function will be a genuine function there are special cases involving semi-transparency where this is not the case. I will not attempt to deal with these special cases here though.

The question thus arises as to how to use the just-defined depth function so as to define a transformation on the metric structure of a sphere, which will account for the size constancy tendency. Inasmuch as the tendency towards size constancy seems to be independent of the direction being looked at Θ, Ф, although as was noted being dependent on the physical depth, p, of the object being looked at, it follows that a visual space will be spherical when objects constituted in it are equidistant from the cyclopean eye, in particular when they are infinitely distant, a case approximated by looking at the sky. If two physical objects at different depths subtend equal solid angles with respect to the eye, the closer object will be constituted smaller in visual space than the more distant one, due to the tendency towards size constancy. Thus, inasmuch as the area on the surface of a sphere projected by a given solid angle is proportional to the square of the radius of the sphere, it follows that a visual space constituted from objects at greater physical depths will possess a greater radius than one constituted from objects at less depth. It also follows that a visual space constituted from objects infinitely distant will possess the greatest possible radius.

The foregoing considerations suggest a possible “external” description of visual space in which the two-dimensional visual space is embedded in a three-dimensional space, and its shape is determined by means of a depth function operating in that three-dimensional space. A mapping can then be defined between the physically-defined spherical polar coordinates, Θ, Ф, and ρ, and a set of spherical polar coordinates Θ’, Ф’, and ρ’, used for the external description of visual space, with the correlations being defined as follows:

Θ’ = Θ [2]

Ф’ = Ф [3]

ρ’ = C – 1/aρ [4]

C is the radius of a visual space in which all of the objects constituted in it are infinitely distant, and a is a proportionality constant. My hypothesis now is that these transformations constitute an external description of the geometry of visual space, where ρ’ gives the distance to a point in visual space in the direction Θ’, Ф’, from a point in a third dimension not in visual space (the center of the sphere in the special case where ρ’ is the same in all directions). It can be noted that ρ’ approaches 0 when aρ approaches 1/C, but inasmuch as there exists a minimum threshold on the depth at which physical objects can be brought into sharp focus, about 6 inches from the eye, this result is consistent with the visual perception process. Also, it is possible that the transformation on the depth function would need to be more complex in order to fully account for the size constancy tendency, but due to unclarity in precisely what the mathematical character of this tendency is, it would seem to be wise to keep the transformation as possible here. It should also be emphasized that p’ has no immediate phenomenal significance, inasmuch as it itself is not contained in visual space, and thus it does not, for example, correspond to perceived visual depth.¹⁰

I will close the paper by briefly mentioning some ways in which the preceding analysis may be able to explain other phenomenal aspects of visual perception. For one point, such an analysis predicts the existence of discontinuities in visual space when there are sharp physical discontinuities in the depth function for relatively close objects, and these discontinuities would seem to be capable of explaining the phenomenon of seeing an “edge” when one relatively close physical object partially occludes the view of another. Another phenomenal characteristic of vision which the analysis seems to be at least in principle capable of explaining is the tendency towards shape constancy; that is, the tendency for the visual shapes of objects seen askew to vary not in accordance with the laws of geometrical perspective, but instead to remain closer to the shapes projected when these objects are seen head-on. The main point ot be made here is that if an object is seen askew, then one edge of it must be closer to the viewer than the other, and thus both ρ and ρ’ will possess different values for these two edges, and the object will be constituted in visual space at a slant also. However, due to the inverse nature of Equation 4 relating ρ’ to ρ, the area of the object as constituted on this slant will not be sufficiently great so as for the object to retain the same shape as when seen head-on, and so it will instead be perceived at a compromise shape in between that shape and the one given by the laws of perspective.

I also believe that in principle my analysis can account for the aspect of phenomenal depth perception which is enhanced by binocular vision. My claim here is that one can apprehend to some extent the internal metric structure of visual space, as for example by noticing the presence of a “corner,” of convexity or concavity, or the present of a discontinuity, in the space. Certainly visual space does not appear to be “flat” when the objects constituted in it possess different physical depths, and I wish to claim that one can use this “lack of flatness,” that is, the apprehension of an internal curvature , as a phenomenal cue for dept. It seems clear that these sorts of phenomenal depth cues are present to some extent even in monocular vision, since one can reverse a “Necker cube,” or see movies in depth using only one eye. As I previously noted, there is also a great deal of evidence that both the tendency towards size constancy and the tendency towards shape constancy are greatly enhanced in binocular vision. Thus, it would seem that the binocular depth cue of retinal disparity enhances monocular phenomenal depth effects by means of increasing the value of the proportionality constant a in the equation relating ρ’ to ρ, and I wish to equate the apprehension of the resulting changes in the internal metric structure of the space with the phenomenal apprehension of visual depth. This may also account for the striking so-called “3D” effects of random stereograms and so-called “3D” movies, although of course I am maintaining that the experiences, even here, are literally still two-dimensional; they just are not flat.

FOOTNOTES

1. See for example Barbara Upton and John Upton, Photography, Third Edition (Boston: Little Brown, 1985), p. 68

2. See Karl Menger, “What is Dimension?” in American Mathematical Monthly 50 (1943), pp. 2-7.

3. Henri Poincaré, Dernieres Pensees, translated b John Bolduc (New York: Dover Publications, 1963).

4. The “cross ratio” of four points is defined as follow:

5. L. P. Clerc, Photography Theory and Practice, edited by D. A. Spencer (New York: Focal Press Ltd., 1970), p. 31

6. Nelson Goodman, Languages of Art (Indianapolis: Hackett Publishing Co., 1976), p. 15.

7. Robert Thouless, “Phenomenal Regression to the Real Object, I,” in British Journal of Psychology 21 (1931), pp. 339-359.

8. A. S. Gilinsky, “The Effect of Attitude upon the Perception of Size,” in American Journal of Psychology 68 (1955), pp. 173-192.

9. On the tendency towards size constancy being enhanced in binocular vision, see Edwin Boring and D. W. Taylor, “Apparent Visual Size as a Function of Distance for Monocular Observers,” in American Journal of Psychology 55 (1942), pp. 102-105; and E. L. Chalmers, Jr., “Monocular Cues in the Perception of Size and Distance,” in American Journal of Psychology 65 (1952), pp. 415-423. On the tendency towards shape constancy being enhanced in binocular vision, see Thouless, “Phenomenal Regression to the Real Object, I I,” in British Journal of Psychology 22 (1931), pp. 1-30.

10. For a description of the resulting internal geometry of visual space see Robert French, “The Geometry of Visual Space,” in Nous 21 (1987), pp. 115-133.

Statement of aims of the SVSGroup

The goal of the multidisciplinary Structure of Visual Space Group is to elucidate through theoretical analysis and empirical research one of the most important aspects of the problem of consciousness that is at present in a state of some confusion, mostly due to errors in reasoning and inadequate conceptualization. We will focus on the inextricably interconnected but specific questions as to the structural nature of visual space—its phenomenology, topology, geometry— what it contains, how it is ‘constructed’, its relation with ‘external’ physical space, and how it is related to Gestalt theory in psychology, correlated brain events, and to corollary theories in modern physics and cosmology (in particular brane theory).
Anyone is free to read the blog. Anyone wishing to post material on the blog is invited to contact either William Rosar (wrosar@ucsd.edu) or John Smythies (smythies@psy.uscd.edu) for information on how to do so. Please attach a short CV when you contact one of us.

This is a Maria Theresa silver thaler whose shape would have given pleasure to Plato

Comments on Jean Nicod’s book “Geometry and Induction”

John Smythies

One of the most important contributors to the subject of the structure and geometry of visual space was the French logician Jean Nicod (1893-1924). The task that he set himself in this famous book was to derive the essentials of geometry, and the justifications for geometry, from our sensations (experiences) of the natural world, rather than from abstract reasoning. Since points do not occur in nature but volumes do, he interestingly based his approach on developments of the volume geometry of Alfred North Whitehead, and not on the point geometry of Euclid (pp 22f).

“ A ‘space’ is “any set of meanings satisfying the axioms of a geometry,,, We ask ourselves whether spaces exist. They do; we have come across one in the domain of numbers. But this does not interest us for we wish to see geometric order reflected, not in ideas but in sensible nature.” (p. 32).

Unfortunately, this book was written in 1922 and was based the naïve realist theory of perception years before the epistemological revolution in neuroscience proved that theory to be wrong (1). It has been established by experiment that we do not perceive the world as it is but as the brain computes it most probably to be (2). This fact has important implications for the logic of visual space.

Nicod claims “The elementary term and relations in nature are sense data. These are what we refer to as this, when we say to ourselves, in speaking of something immediately present to one of our senses, this is a tree, this is a penny, or again, this is a shooting star, this is the song of a nightingale.”(p.35)

However, what is “immediately present to one of our senses” is a physical object not a sense-datum. Secondly, we do not experience these physical objects directly: that which we experience is the phenomenal image of the external object created by the brain’s neurocomputational mechanisms. It is legitimate to base a geometry of the cosmos on the arrangement of the physical objects in that cosmos that we perceive via our sensations. It is also legitimate to base a geometry of our phenomenal world on the arrangement of the (usually visual) sensations that we experience by introspection. But it is not legitimate to confuse these two processes and attempt to base the geometry of the cosmos on the order of our sensations: nor is it legitimate to try to base the geometry of our sensations on the geometry of the external objects that these sensations represent. To help avoid this confusion we should not use the terms “sense-datum” and “sensum” (that derive from Russell, Broad and Price), as this tends to lead to confusion between object and representation: we should use the standard neurological term ‘sensation’ instead. Compare Nicod’s correct usage in—

“But at present I do not seek the most economical reconstruction of the order of the flux of my sensations. I am simply inspecting the relations I discover there. I try to apprehend each one of them as it presents itself to my mind.” (pp. 55-56).

Here he is clearly talking about his sensations.

with his incorrect usage in—

“The mind whose existence we are assuming is thus confronted with an infinity of distinct and simultaneous sense-data, corresponding term for term to the points in our physical space. This is the greatest possible intuitive perception of space.” (p. 107)

note also his statement

“Such are the elementary terms and relations of the sensible process. We have merely sought to present them before the mind’s eye. (p 66)

Note here also how Nicod says our sensations are presented to one’s “I” or to one’s “mind” or “mind’s eye”. This usage is currently unfashionable, but I think correct.

___________________________________

Nicod also lists some fundamental topological properties characteristic of the events that occur in a person’s field of vision (sic)—

‘Interiority”

He gives as an example the flight of an eagle seen by a person. During the eagle’s majestic glide it gives one flap of its huge wings—

Nicod says that the event ‘flap’ is interior to the flight.

However, here he is clearly primarily talking about the eagle as a physical object, not about his sensations of the eagle. (p. 37). The terms “field of vision” (or “stimulus field”) commonly refer to (A) the collection of external physical objects in range of vision at that moment. The term “visual field” refers to something quite different—to (B) the collection of visual sensations that we experience in phenomenal consciousness: B represents A.

Nicod says that interiority may also be experienced in somatic terms—e.g. by running a pencil over different fingers of one’s hand.

“This relation is distinct and obvious and springs to the [mind’s I hope] eye in each case.”

___________________

Thus ‘Interiority’ has both durational and extensional points of view. The squares of the physical chessboard are spatially internal to the rim of the physical board in physical space. Furthermore the squares that make up the phenomenal visual sensation of the chessboard in the visual field in consciousness are spatially internal to the visual image of the rim of the chess board: whereas the flap of the eagle’s wing isspatio-temporally internal to its flight in space-time (differentiated as above between physical space-time and phenomenal space-time). Nicod tends to confuse physical space and phenomenal space.

______________________

Nicod adds other basic topological relations in the same family as ‘interiority’——

‘Interpenetration’ (e,g, a row and a column in chess board)

‘Exteriority’ (e,g, 2 rows on chess board)

‘Continuum’ (touch but not penetrate: e.g. 2 runners in a relay race)

and equivalent temporal relations

‘Temporal inclusion’ (during)

‘Interference’ (overlapping)

‘Separation’ (not overlapping)

‘Prolongation’ (one starts as the other ends)

These same relations also apply to the visual field but in the different context of observable relations experienced between sensations rather than observable relations seen (perceived) between external physical objects.

Moreover, Nicod leaves out a large number of basic topological relations that are to be found in visual space (see further communications).

References

1. Smythies J.R. Ramachandran VS. 1998, An empirical refutation of the Direct Realist theory of perception. Inquiry, 40, 437-438.

2. Smythies J. 2009, "Philosophy, perception, and neuroscience" Perception 38(5) 638 – 651.