Spectrum Inversion and the Color Solid

Southern Journal of Philosophy, 23 (4), Winter 1985, 431-443.

Austen Clark
Department of Philosophy U-54
University of Connecticut
Storrs, CT 06269-2054

The possibility that what looks red to me may look green to you has traditionally been known as "spectrum inversion." This possibility is thought to create difficulties for any attempt to define mental states in terms of behavioral dispositions or functional roles. If spectrum inversion is possible, then it seems that two perceptual states may have identical functional antecedents and effects yet differ in their qualitative content. In that case the qualitative character of the states could not be functionally defined.

Of course our initial characterization of spectrum inversion does not provide such a counterexample. If the only difference between Jack and Jill is that what looks red to Jack looks green to Jill, then that difference could immediately be detected in their different behaviors and reports. For example, Jack could discriminate (what he calls) red from green things, while Jill could not. Even if we add the supposition that what looks green to Jack looks red to Jill, there would still be behavioral and functional differences between them, in that Jack would find the color of oranges (for example) more similar to that of apples than to limes, while Jill would not. For spectrum inversion to provide a counterexample to functionalism, the "switch" in colors between Jack and Jill must somehow be systematic. This paper is concerned with analyzing the exact sense in which the switch must be systematic.

As a start we can specify three conditions which are necessary for spectrum inversion to show that the qualitative contents of color sensations cannot be functionally defined. I shall use "Jack" and "Jill" as names for the prototypical invert pair. (1) Jack and Jill must have coextensive color discriminations and judgments, so that for any two stimuli x and y, Jack finds x indiscriminable from y if and only if Jill does as well. Jack and Jill must also make precisely the same judgments of relative similarity, so that Jack judges x to be more similar to y than to z if and only if Jill does as well. (2) Jack and Jill must be functionally isomorphic, so that color sensations play functionally equivalent roles within the psychology of each. <Note 1> (3) There is some mapping of color stimuli onto qualia, which is consistent with (1) and (2), and which is such that the color quale Jack has when viewing some color chip z is not qualitatively the same as the one Jill has when viewing z. Instead Jack's quale is the same as one Jill has when viewing some distinct y. < Note 2>

Cases which fail to meet any of these three conditions are no counterexample to functionalism. What makes spectrum inversion a counterexample to functionalism is precisely that Jack's sensation of some color chip z is qualitatively different from that of Jill's, even though those sensations are functionally and behaviorally equivalent. Functionalism has no difficulties with a case which fails to meet condition (1), for if the discriminations of Jack and Jill are not coextensive, then there is presumably some difference in the causal role subserved by their respective color sensations, and the case fails to demonstrate qualitative differences conjoined with identical functions. Condition (2) is necessary to cover aspects of the causal role of color sensations beyond their purely behavioral effects (e.g., their relations to other sensations, beliefs about colors of objects, emotional associations, synesthesia, memories of colors, imagery, and so on).

Interestingly, the need for coextensive discriminations (condition (1)) shows that the the supposition of spectrum inversion is contrary to fact. Between any two individuals there is likely to be some part of the spectrum for which one of the individuals has better discriminations than the other, so that there are at least two colors x and y such that one individual can discriminate x from y while the other cannot. <Note 3> It would follow that those two individuals do not see exactly the same colors. This of course does not demonstrate spectrum inversion, since it merely demonstrates differences in the causal role of color sensations in different people (instead of the required identity in causal roles, conjoined with qualitative differences). The latter case is hypothetical, but a hypothetical case will suffice: the mere logical possibility of spectrum inversion would show color qualia cannot be functionally defined.

The key question is: if (1) and (2) are satisfied, could there be a mapping as described in (3)? What conditions must it satisfy?

I. Conditions on a Mapping Function

Suppose we model judgments of similarity by distances between points, so that if stimulus x is judged more similar to y than to z, then the point corresponding to x is closer to y than to z. Furthermore, two points will be distinct if the color stimuli corresponding to them do not match the same sets of points. Finally, if u and v are relatively as similar as x and y, then the distance between u and v should be the same as the distance between x and y. Under these conditions, colors form a three dimensional solid, with axes (for example) of hue, saturation, and brightness. Equal distances in the solid should correspond to equivalent degrees of similarity between the corresponding colors. <Note 4> The axes of the solid can be redefined in terms of the proportions of particular reference wavelengths (or standard primaries) required to produce a color match for that point, and one can then view the color solid as an ordering of colors in terms of particular physical characteristics of stimuli. The facts of color mixing and matching can be represented perspicuously by addition of vectors in such a space. There are various systems for naming and ordering the result; one is called the "Munsell color solid" which comes complete with maps of each plane and several hundred color samples.

The color solid can be used to describe conditions on the mapping function required in spectrum inversion. If all judgments of relative similarity and discriminability are represented in the solid, then any mapping function must be consistent with that solid. In order to map a given stimulus onto two distinct color qualia in Jack and Jill, it seems that one must be able to map colors onto distinct colors. For this to preserve functional equivalence, it seems the mapping must be systematic, inverting the entire color solid. If for some reason this cannot be done--if, for example, the metrics and topology of the solid show that it cannot be inverted--then on this line it would follow that spectrum inversion is impossible, and that no mapping function is consistent with our psychological makeup. <Note 5> Although it is peripheral to his main argument, Sydney Shoemaker seems to advance such a claim. He says:

Taken one way, the claim that spectrum inversion is possible implies a claim that may, for all I know, be empirically false, namely that there is a way of mapping determinate shades of color onto determinate shades of color which is such that (1) every determinate shade (including 'muddy' and unsaturated colors as well as the pure spectral colors) is mapped onto some determinate shade, (2) at least some of the shades are mapped onto shades other than themselves, (3) the mapping preserves, for any normally sighted person, all of the 'distance' and 'betweenness' relationships between the colors (so that if shades a, b, and c are mapped onto shades d, e, and f, respectively, then a normally sighted person will make the same judgments of comparative similarity about a in relation to b and c as about d in relation to e and f), and (4) the mapping preserves all of our intuitions, except those that are empirically conditioned by knowledge of the mixing properties of pigments and the like, about which shades are 'pure' colors and which have other colors 'in' them... <Note 6>

Shoemaker immediately points out that facts concerning our psychological makeup are irrelevant to determining the logical possibility of spectrum inversion, and that it is only the logical possibility which is of philosophical importance. With that we can immediately concur; the question here however is whether spectrum inversion implies any facts concerning our psychological makeup. Could any facts concerning the color solid show that inversion does not occur?

Shoemaker's points (3) and (4) fill out some conditions necessary for a mapping function to be consistent with discriminal and functional equivalence. For reasons which will soon be apparent, I shall call them "symmetry conditions." Need they be added to our original list of requirements for spectrum inversion? We should first determine what characteristics of the color solid they imply.

In order to preserve isomorphism of discriminations, a mapping function must preserve all the distance relationships among colors. If a color x is near another color y, then no matter what new stimuli lead to x and to y under spectrum inversion (and so no matter what their new locations are), those colors must still be near (relatively similar to) one another. Wherever a point moves, nearby (and similar) points must move as well. If not, similarity judgments will differ between normal and inverted conditions. If in Jack x is shifted without its neighbors, then his similarity judgments will change, as thereafter x will seem to him similar to some new z, and dissimilar from its old neighbors.

Colors must move as entire neighborhoods; and clearly this condition generalizes to the entire solid. A consequence of this is that if the color solid is invertible, then it must in a certain way be symmetric.

What kind of symmetry is required? We need a mapping in which at least one point is mapped to some different point, and in which if the distance from x to y is d, then the distance from the inverse of x to the inverse of y is d as well. This implies that there must be at least one point identical to its inverse--so that the solid is symmetrical in some way. (If at least one point moves, then at least one must not move.) For suppose there were no such point, so that every point is mapped to some distinct inverse. The distances between points and their inverses will vary; pick the point e which is closest to its inverse e' among any of the point-inverse pairs. Now pick a point f between e and e'. Its inverse must also be between e and e', and so we get the contradiction that the f-f ' distance is less than the e-e' distance.

If the solid is not continuous but discrete then there is no guarantee that there will be a color at point f. Nevertheless the conclusion still follows, for there must be an axis of symmetry through point f, even if no color corresponds to that location.

The solid may be symmetric with respect to a plane, a line, or a point. The plane, line, or point provides what may be called a "symmetry locus." Each point in the solid has some shortest possible projection line to the symmetry locus, and in each inversion is mapped to that point equidistant from the locus along the same projection line. There are as many distinct inversion mappings for the solid as it has distinct symmetry loci. The solid may be symmetric with respect only to one plane, as is the typical coffee mug, which could only be inverted in one way. It may be symmetric with respect only to one line, as are some pairs of scissors, which again would have only one inversion. Finally, it may be symmetric with respect only to one point, as are left and right hands placed in opposite directions on opposite hemispheres of a sphere. Combinations of symmetries yield other possibilities. The solid may be symmetric with respect to two planes and a line, but not a point (as is a ping pong paddle, with three inversions); to a line and an infinite number of planes (a wineglass), or to a point and an infinite number of lines and planes (a sphere).

If the color solid were some odd shape (such as a right hand glove) which is not symmetric with respect to any point, line, or plane, then (according to the symmetry conditions) no mapping of colors could preserve all judgments of relative similarity. There would be no way to shift any of the colors without altering at least some relationships of relative proximity. If the point corresponding to red were located on the thumb nail, for instance, there is no other location on the glove to which one could map red while preserving all of its distance relationships to neighbors: there is no other place on the glove "shaped" quite like the thumb nail. So an asymmetric shape will not be invertible if one preserves all judgments of relative similarity of color.

The symmetry conditions seem to follow from the need to preserve ordinary judgments of color similarity. The problem with the hand-shaped color solid is that there seems to be no mapping of colors to new locations in such a way as to preserve our judgments of similarities of color.

Interestingly, the symmetry conditions would (again) show the supposition of spectrum inversion to be contrary to fact, in that the human color solid as a matter of empirical fact is asymmetric with respect to every point, line, and plane, and cannot be inverted. The Munsell color solid is an odd shape with an uneven bulge in the purples and reds; there is no corresponding bulge on the other side. <Note 7> Humans are best at discriminating saturation differences at the ends of the spectrum, among the reds and blues. In effect the "density" of discriminable points in those regions cannot be matched anywhere else in the solid, and that is why the solid is distended there, and so cannot be inverted. <Note 8> Mapping those points to any other place would change some of their distance relationships.

Asymmetry can be shown in a second way. For the symmetry conditions to be met, certain lines running through the color solid must map onto themselves under inversion. One is the line along which humans can make the greatest number of color discriminations; points on that line must be mapped to other points on that line. This maximal discrimination line seems to run through the achromatic "white" axis of the solid to locations corresponding to purple. A second line is that along which humans can make the fewest number of discriminations; it runs from the "white" axis to regions corresponding to yellow. These lines intersect one another at the white axis of the solid. Since each line must be mapped onto itself, the solid can be inverted only if they intersect at the midpoint of one (or both) of the lines. But the achromatic white axis of the solid lies at the midpoint of neither line. <Note 9> Hence the human color solid cannot be inverted. <Note 10>

An interesting implication of the asymmetry of the actual human color solid is that a structural definite description employing no singular terms can be used to precisely identify every color. That is, there are "landmarks" within the color solid which can be identified purely relationally, and the location of each color within the solid can be uniquely identified in terms of its relations to those landmarks. Every color name could be replaced by a definite description of the form "the x bearing R to z such that..." where R is just the relation "indiscriminability." Such a definite description is structural in that it identifies colors solely by their place within the pair list of indiscriminabilities--e.g., by their structural properties. <Note 11>

II. Relative Similarity among Qualia

At this point one may justifiably suspect that something has gone awry in the formulation of the symmetry conditions. Why is the structure of the color solid (and hence indirectly of all the discriminations and judgments of similarities and differences among colors) even relevant to the problem of qualia inversion? All of the empirical considerations concerning distances in the color solid, purple bulges, maximal discrimination lines, and so on seem beside the point when it comes to the suggestion that our color sensations differ qualitatively. Do we really presuppose all that when we speculate that what looks red to me might look green to you?

If we do, then the asymmetry of the color solid, while failing to settle the conceptual question of qualia and functionalism, does provide evidence that inversion does not occur in the actual world. This implication seems unacceptable. I shall argue that the symmetry conditions are unnecessary, and that inversion implies no facts concerning the psychological color solid. A mapping function can be consistent with discriminal and functional equivalence without satisfying symmetry conditions. Instead the mapping function needs to take into account the structure of similarities among color sensations (or qualia) which form a distinct domain from that of color stimuli. So the asymmetry of the human color solid does not demonstrate that inversion fails to occur in the actual world. To show this we should reconsider the relation between the color solid and inversion.

In some ways "spectrum inversion" is an unfortunate misnomer for the mapping problem, since colors are ordered three dimensionally, not one-dimensionally. How is spectrum inversion represented in the color solid?

The obvious answer--that one systematically switch the color chips attached to points in the solid--is easily seen to be incorrect. No such switch is required or allowed under the conditions of spectrum inversion. One solid, with chips attached in the standard way, will do for everyone--even if some viewers are presumed to have inverted qualia relative to others.

A given chip in the solid is printed so as to present to the viewer a stimulus which is indiscriminable from the particular mixture of standard primaries [x, y, z] corresponding to that point. No matter what quale it engenders, the standard chip at that point does the job correctly for everyone. The chip may present Jack with a red quale and Jill with a green one, but nevertheless gives for each viewer a stimulus indiscriminable from the mixture of primaries [x, y, z]. (In fact it will be indiscriminable from any stimulus whose wavelength components sum to the vector [x, y, z].) <Note 12>

Any change in the solid would violate a condition of qualia inversion. Suppose the standard solid suffices for Jill, but (at least) one chip C is moved in it to represent the qualia of Jack. Then Jill and Jack cannot have coextensive discriminations. Let M be a stimulus composed of the mixture of standard primaries associated with the coordinates [x, y, z] of C in Jill. Then Jill will find C and M indiscriminable, while Jack will find them discriminable. Since Jack and Jill differ in their discrimination behaviors, a pre-condition of spectrum inversion is not met. So the color solid is accurate for everyone, even under conditions of qualia inversion, and not a single chip in it should be moved. <Note 13>

Spectrum inversion has so often been associated with a switch in the colors that this conclusion may come as some surprise. What switches if not the chips?

One needs to switch, not the ordering of stimuli as presented in the color solid, but rather the the qualitative character of the sensations associated with those stimuli. The same set of stimuli will be judged indiscriminable from each mixture of standard primaries [x, y, z], but the quale associated with each set will be different. The color solid represents the structure of discriminations among color stimuli, not the qualia associated with those stimuli.

In other words, we need to dissociate color qualia from the color chips, and imagine a reallocation of the former, not the latter. Each chip stays where it is, but its color (now representing a quale) is to change.

III. Qualitative Isomorphism

Once one sees that color qualia (and not colors) are inverted, the relevance of the structure of the color solid becomes much more problematic. Relationships of similarity among colors do not necessarily define relationships of similarity among color qualia. The color solid constrains the former, but not necessarily the latter.

Qualia inversion in a totally asymmetric solid is possible if the relation of relative similarity among color qualia does not behave in the same way as relative similarity among colors. Imagine the color solid drained of all color, and partitioned into regions of equal-interval color change. Each volume is defined solely by its boundaries with neighbors and position in the overall solid. Is it possible to find two distinct assignments of qualia to that structure which both preserve discriminal and functional equivalence? In fact any assignment of different qualia to different regions is possible, as long as we suppose that relative similarities among qualia satisfy the structure given by distances between points. This is possible even if colors (as a matter of empirical fact) can only satisfy that structure in one way.

Granted, in judging similarity of stimuli, yellow is closer to orange and to green than it is to blue, but there is no guarantee or requirement that similarities among qualia proceed analogously. We simply suppose that each person is built in such a way that the quale associated with any given point in the color solid seems most like those qualia (whatever they are) that are associated with neighboring points. Perhaps Jack has a visual-psychological organization such that for him the qualitative content of sensations of that certain shade of yellow is more similar to the qualitative content of sensations of blue than to that of other yellows. We require a isomorphism, not between two color solids (one the inverse of the other), but rather between the color solid and the qualia solid. Chips must be mapped onto qualia in such a way that whenever chip x is closer to y than to z, the quale occasioned by x is relatively more similar to the quale occasioned by y than to that by z. I will call this condition "qualitative isomorphism."

Given qualitative isomorphism, any mapping function is consistent with discriminal and functional equivalence. While the color solid shows no obvious borders, and appears continuous, the equivalent structure for qualia could be a gaudy jumble (with a bright green next to pink, bordered with orange, and so on), as long as the mechanisms for judging qualitative similarity are such that the "pink" quale is judged relatively more similar to the nearby green one than to the more distant orange. Given that condition, discriminations and judgments of color stimuli will be undisturbed. So qualitative isomorphism is a sufficient condition on mapping functions.

Furthermore, qualitative isomorphism does not require symmetry in the color solid. A simple example shows this conclusively: use modalities other than color as a model for the qualitative contents of color sensations. It is imaginable that the human sense of smell be much better articulated than it is, perhaps approaching canine sensitivity. Suppose that it orders odors in three dimensions, and that the structure of relative similarities among smells is isomorphic to the color solid. Perhaps Jack has olfactory qualia attached to color stimuli, while Jill does not. Such a case would nevertheless constitute "spectrum" inversion, in that Jack and Jill would have qualitatively distinct sensations even though they are discriminally and functionally isomorphic. The asymmetry of the color solid implies nothing concerning smell qualia, and so does not rule out inversion.

The color solid defines a structure of relationships among stimuli, in which relative proximity corresponds to relative similarity. That structure can be instantiated by various sets of properties. It may have different models in different people. As long as discriminatory mechanisms are built in such a way that neighboring states yield judgments of a high degree of similarity, any set of properties may underlie such discriminations.

In effect the idea of inverting, rotating, or mapping the color solid is an analogy for the possibility that different properties underlie color discrimination in different people. Given the structure of discriminations and relative similarities, one notices that any set of properties could be associated with a given point, as long as the various discriminations and similarity relationships are retained. One must establish a certain structure of proximities, but beyond that, any set of properties will do. The structure of properties which springs most readily to mind is precisely that of color: one supposes that that structure is preserved under rotations, inversions, and so on. While this may or may not be possible, the important point is that differences in color serve merely as an analogy for the differences in properties perhaps subserving discriminations in different people.

Why then is the structure of similarities among color sensations confused with the structure of similarities among colors? One reason we confuse the two structures is that the patchwork quilt of multicolored qualia is strongly counterintuitive. Even though sensations are not literally pink or green (so need not obey rules of similarity for colors), it is difficult to imagine how a sensation of pink could be qualitatively more similar to a sensation of green than one of red. We have such difficulties precisely because the intuitive model for color qualia (or, in Sellarsian terms, color "impressions") are colors themselves, <Note 14> and, on many accounts, identity criteria for qualia can be defined in terms of relationships of similarity and dissimilarity ("matching" or "indiscriminability") among their occasioning stimuli. Take Goodman's classic treatment. On his account, if two stimuli occasion identical qualia, then clearly enough those stimuli must match one another. Further, if two stimuli both match just the same sets of stimuli, then there is no qualitative difference between them, and they present identical qualia. <Note 15> So necessary and sufficient conditions for identity of qualia can be framed in terms of matching conditions among stimuli. Those matching conditions provide the stimulus with a coordinate in the color solid, and so identity conditions for qualia can in effect be defined in terms of locations within the color solid. If such identity criteria for qualia are accepted, then each point in the color solid yields a unique quale, and both our gaudy qualia solid and qualia inversion (given asymmetry of the color solid) are ruled out of the actual world.

IV. Relations and Locations

Whatever the definition of qualitative similarity, it is clear that the metrics and topology of the color solid give no grounds for ruling out the possibility that different properties instantiate color sensations in different people. What are the implications of this admission for functionalism? Multiple instantiation is, after all, a functionalist platitude. The functionalist admits that different sets of properties could satisfy the relational structure defined by the color solid. Any set of properties will do, as long as they stand in the appropriate pattern of relationships.

But there are two ways to construe this possibility. One is to urge that since the instantiating properties differ across people, qualitative contents do as well. On this line, even if two stimuli are shown to have the same respective locations in the color solids of two people, the qualia occasioned by those stimuli may differ. The second response is to eschew the identification of qualitative content with instantiating properties, retain purely relational criteria for qualitative similarity, and accept multiple instantiations of a given quale. On this line, once two points are shown to have the same respective location in the color solids of two individuals, there is no clear sense to the further suggestion that their associated sensations are somehow qualitatively dissimilar.

Which alternative is more justified? The critical issue is whether there is a clear concept of qualitative similarity which applies across the different properties (possibly) instantiating the color solid in different people. Why should differences in instantiating properties across individuals have any claim to the label "different qualitative content"? We require some clear sense in which a sensation of pink (e.g., of that location in the color solid) in Jill may be qualitatively identical to one of green (a distinct location) in Jack.

This is difficult to provide, since (as noted above) the concepts of qualitative similarity and qualitative identity are typically tied to precisely those considerations of discrimination, matching, and judgment which define location in the color solid. Take your own putatively private qualia: even among them, if x does not match y, then x and y do not present identical qualia; and if they do present identical qualia, then there is no z which matches one but not the other. What sense of similarity could overrule such appeals to discrimination? There may be such a sense, but it has never clearly been articulated. Friends of qualia instead here tend to appeal to intuitions, particularly those concerning various conveniently counterfactual situations.

While argument cannot be expected to alter intuitions, an analogy may make those of the functionalist more plausible. Qualia identification is in many ways analogous to identification of places. <Note 16> (Indeed, for a functionalist, qualia identification is a sort of spatial identification, where the "space" is that defined by the order yielded by qualitative similarities and dissimilarities.) Goodman articulated a position with which the functionalist can agree:

On the basis of all such information at our command, we construct a map that assigns a position to each of the described qualia. Quale names may then be treated as indicating positions on this map. Indeed, to order a category of qualia amounts to defining a set of quale names in terms of relative position, and thus eventually (in our system) in terms of matching. When we ask what color a presentation has, we are asking what the name of the color is; and this is to ask what position it has in the order--or in other words to ask which of the ordered qualia it matches. <Note 17>

To identify a particular shade, one typically either presents a sample or describes its relations to samples with which one's interlocutor is already familiar. Similarly, to identify a location one either gives a proper name of a place with which one's interlocutor is familiar, or a set of spatial relationships from some collection of such places (e.g., north of fifteenth street, two blocks west of the bank, etc.). Many different coordinate schemes could be used to give an answer. As long as spatial relations among objects are accurately represented, any unit of measurement, direction, and origin would work, satisfying all navigational needs. Relative to a system of coordinates, description of location makes sense; yet multiple coordinate schemes are possible.

Corresponding to spectrum inversion is spatial inversion. One can imagine a universe which is spatially symmetric, so that relative to some locus every object has an exact spatial inverse. (On the far side of the locus, there is a Twin Earth, Twin Venus, and so on.) In such a universe, two places would satisfy the same spatial structural description, even when the spatial relations in which each stands are extended to include every object in the universe. If the coordinate scheme is appropriately symmetric, every n-tuple of coordinates may be ambiguous in this way. But such a possibility fails to show that spatial location is not relational--that one can identify some "absolute" location independently of the network of spatial relationships in which it stands. <Note 18> Position is defined by relationships within some coordinate scheme. If ambiguity persists, we can disambiguate by ostension. Place names satisfying exactly the same pattern of spatial relations to some ostended place (typically, the speaker's location) name the same place. There is no further fact of the matter which would show their reference differs. Even in a symmetric universe, a speaker has but one place at a time, and Earth can be differentiated from its twin by ostension to the speaker's location. Although they may both satisfy the same spatial structural description, the speaker on Earth and his Twin Earth doppelganger can each refer to their own home planet with the humble demonstrative "here."

Similarly, even if the structure of qualitative similarities is symmetric, locations in it (e.g., of a particular quale) can be disambiguated by ostension to samples. To define "green qualitative content," first use conditions of matching to define the relationship of qualitative similarity, then point to an exemplar. Green qualitative contents are those qualitatively similar to the one occasioned by that (pointing). This identification can succeed for everyone, even under the supposition of spectrum inversion. <Note 19> The friend of qualia may question whether such a description picks out the same qualitative content in everyone. Here too the response is that once such relations are fixed, there is no further fact of the matter; the respective sensations satisfy all the determinate senses of qualitative similarity, and no other sense has been provided. To further query identity of qualia is analogous to querying identity of location, once spatial relations to some ostended place have been provided.

Just as with spatial location, therefore, the functionalist holds to a relational theory of qualitative content. To specify the content of the sensory state is just to specify its place in a network of relationships of relative similarity and discriminability. To query its qualitative content in any more absolute way is as meaningless as querying location after coordinates have been given.

Note that for the functionalist, the concept of a quale is the concept of a place within the color solid, but it is not the concept of whatever property in us happens to instantiate that place. Similarly, a spatial location is distinct from whatever object happens to occupy it. Granted, in our color discrimination mechanisms there presumably is some physiological property which we use to recognize that place; that is, which satisfies the given pattern of relations within the overall network of similarities and differences among stimuli. Such a property will "instantiate" or "realize" the quale. However, the concept of a quale is not the concept of that property, but rather the concept of the place it occupies. <Note 20> What makes that property instantiate that quale (what allows us to use it to recognize that qualitative content) is just that it have the appropriate location, not that it be the physiological property that it is. A different physiological property in the same respective place in the color solid would realize the same quale.

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1. There are various precise formulations of the meaning of "functional equivalence." See Ned Block, "Introduction: What is Functionalism?" and "Troubles with Functionalism," both in Ned Block (ed.) Readings in the Philosophy of Psychology, vol. 1 (Cambridge Massachusetts: Harvard University Press, 1980), pp. 171-184, 268-305. <Back>

2. As Ned Block puts it, "the sensation that you have when you look at red things is phenomenally the same as the sensation that I have when I look at green things." See "Are Absent Qualia Impossible?" Philosophical Review, LXXXIX, 2 (April 1980), p. 257. <Back>

3. See Robert M. Boynton, Human Color Vision (New York: Holt Rinehart and Winston, 1979), pp. 338-340; Gunter Wyszecki and W. S. Stiles, Color Science (New York: John Wiley, 1967), pp. 419-421, 425-435. <Back>

4. A representation of colors which meets this condition is called a "uniform color scale." While there are several usable approximations to such a scale, in fact no single system exists which has managed to fit all available data and satisfy all color scientists. Indeed, one difficulty is that there seems to be no way to arrange points (representing colors) in a three dimensional Euclidean space in such a way that equal distances correspond everywhere to equal color differences. Instead it seems the best fit for the geometry of the color solid is Riemannian. No matter how coordinate schemes are assigned or transformed, the human color space has a persisting nonzero Gaussian curvature. Color geodesics for such a space (representing the smallest color differences between points) are invariably "curved." See Boynton, Human Color Vision, pp. 280-282, and Wyszecki and Stiles, Color Science, pp. 518-525, 527, 533. The latter authors conclude that "the geometry of color-perception space is not Euclidean." (ibid., p. 453). <Back>

5. Precisely such an argument is found in Bernard Harrison, Form and Content (Oxford: Basil Blackwell, 1973), pp. 102-115. <Back>

6. Sydney Shoemaker, "Functionalism and Qualia," Philosophical Studies 27 (1975), pp. 301-302. See also Sydney Shoemaker, "The Inverted Spectrum," Journal of Philosophy, LXXIX, 7 (July 1982), p. 366. <Back>

7. See Wyszecki and Stiles, Color Science, p. 477. <Back>

8. See Robert W. Burnham, Randall M. Hanes, and C. James Bartleson, Color: A Guide to Basic Facts and Concepts (New York: John Wiley & Sons, Inc., 1963), pp. 161, 139. <Back>

9. See ibid., figure 6.8, p. 139. <Back>

10. One may object here that the problems so far mentioned are purely metrical, having to do with incompatible distance measurements. Such measurements are a function of the coordinate scheme chosen, and perhaps, the objection goes, one could provide some transformed coordinate scheme for the color solid which would smooth its asymmetric bulges and eliminate incompatibilities among distances. This objection fails, however, since in a uniform color scale (see note 4) relationships between distances are fixed, and so no transformation of coordinates can be allowed to change its shape. But there may also be a purely topological barrier to inverting the human color solid. As mentioned above, the geometry of the solid is Riemannian, with nonzero Gaussian curvature. Gaussian curvature is a topological invariant of a space, independent of whatever coordinate scheme is used. While different studies have provided different estimates of the curvature of human color space, they agree that it varies from point to point in an asymmetric fashion. Such a nonuniform curvature forestalls inversion, no matter how the solid is coordinatized. See Wyszecki and Stiles, Color Science, pp. 522-523, 531-533. <Back>

11. This possibility is mentioned in Rudolf Carnap, The Logical Structure of the World (Berkeley: University of California Press, 1967) pp. 24-28, as well as in Nelson Goodman, The Structure of Appearance, 3rd edition (Boston: Dordrecht Reidel, 1977), p. 244. <Back>

12. An infinite number of different combinations of wavelengths may yield stimuli indiscriminable from point [x, y, z], provided that each sums (vector-wise) to that point. See Tom Cornsweet, Visual Perception (New York: Academic Press, 1970), p. 192. <Back>

13. See Austen Clark, Qualia and the Psychophysiological Explanation of Color Perception, Synthese, forthcoming. <Back>

14. See Wilfrid Sellars, Science, Perception, and Reality, (London, Routledge & Kegan Paul, 1963) pp. 191-194. <Back>

15. Goodman, The Structure of Appearance, pp. 196, 209-210. See also Bernard Linsky, "Phenomenal Qualities and the Identity of Indistinguishables," Synthese 59 (1984): pp. 363-380. <Back>

16. This analogy was suggested by some remarks of W. V. O. Quine in Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), pp. 49-51. <Back>

17. Goodman, The Structure of Appearance, pp. 200-201. <Back>

18. This analogy is not meant to invoke the dispute between relational and substantival theories of space-time. Instead it trades on the neutral claim that identity of spatial location can be defined in terms of spatial relationships. (The substantivalist will allow spatial relations to obtain between unoccupied space-time points, while the relationist will not; but with that proviso both can accept this claim.) To identify a location, it suffices to describe such spatial relationships; there is no further "fact" of the matter as to where something is located. <Back>

19. This follows from the argument in section 2 that the standard placement of chips in the color solid represents the perceptions of everyone, even under the supposition of spectrum inversion. <Back>

20. This is in some ways close to the view of Paul and Patricia Churchland, "Functionalism, Qualia, and Intentionality," Philosophical Topics, 12, 1 (Spring 1981), pp. 121-132; and Paul Churchland, Matter and Consciousness (Cambridge, Massachusetts: MIT Press, 1984), pp. 38-42, 52-53, 60. <Back>

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