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Color, Consciousness, and the Isomorphism Constraint

Abstract

The relations among consciousness, brain, behavior, and scientific explanation are explored within the domain of color perception. Current scientific knowledge about color similarity, color composition, dimensional structure, unique colors, and color categories is used to assess Locke's "inverted spectrum argument" about the undetectability of color transformations. A symmetry analysis of color space shows that the literal interpretation of this argument -- reversing the experience of a rainbow -- would not work. Three other color-to-color transformations might, however, depending on the relevance of certain color categories. The approach is then generalized to examine behavioral detection of arbitrary differences in color experiences, leading to the formulation of a principled distinction, called the isomorphism constraint, between what can and cannot be determined about the nature of color experience by objective behavioral means. Finally, the prospects for achieving a biologically based explanation of color experience below the level of isomorphism are considered in light of the limitations of behavioral methods. Within-subject designs using biological interventions hold the greatest promise for scientific progress on consciousness, but objective knowledge of another person's experience appears impossible. The implications of these arguments for functionalism are discussed.

In this article I discuss the relations among mind, brain, behavior, and science in the particular domain of color perception. My reasons for approaching these difficult issues from the perspective of color experience are two-fold. First, there is long philosophical tradition of debating the nature of internal experiences of color, dating from John Locke's (1690) discussion of the so-called "inverted spectrum argument". This intuitively compelling argument constitutes an important historical backdrop for much of the article. Second, color is perhaps the most tractable, best understood aspect of mental life from a scientific standpoint. It demonstrates better than any other topic how a mental phenomenon can be more fully understood by integrating knowledge from many different disciplines (Kay & McDaniel, 1978; Thompson, 1995; Palmer, in press). In this article I turn once more to color for new insights into how conscious experience can be studied and understood scientifically.

I begin with a brief description of the inverted spectrum problem as posed in classical philosophical terms. I then discuss how empirical constraints on the answer can be brought to bear in terms of the structure of human color experience as it is currently understood scientifically. This discussion ultimately leads to a principled distinction, called the isomorphism constraint, between what can and what cannot be determined about the nature of experience by objective behavioral means. Finally, I consider the prospects for achieving a biologically based explanation of color experience, ending with some speculations about limitations on what science can achieve with respect to understanding color experience and other forms of consciousness.

1 DETECTING TRANSFORMATIONS OF COLOR EXPERIENCE

The basic intuition that underlies Locke's inverted spectrum argument is that other people might have the same overall set of color experiences as you do but differently connected to objects in the external world. When you and I look at the same red apple under the same lighting conditions, for example, do we have the same internal experience of redness, or might I have the experience you call greenness, or yellowness, or whatever? The issue is perhaps most clearly captured by the situation of looking at a rainbow. You perceive a particular ordering of chromatic experiences from red at the top to violet at the bottom, but I might perceive exactly the reverse of your experiences, with violet at the top and red at the bottom. We would both name the color at the top "red" and the one at the bottom "violet," of course, because that is what we all have been taught. Color naming is presumably mediated by internal color experiences, but it is only the sociolinguistically sanctioned stimulus-response associations that matter for our ability to communicate about colors. The fact that we name the same objects and lights with the same color terms therefore is not sufficient to determine whether or not our internal chromatic experiences are the same.

The rainbow-reversal interpretation of the inverted spectrum argument is quite literal in the sense that we have supposed that my experiences of the spectrally pure (monochromatic) colors are simply the "inverse" of yours about the spectral midpoint. Notice that the "inverted spectrum argument" is actually something of a misnomer: It is not the spectrum that is inverted -- i.e., nothing has happened to the rainbow itself -- but the inner experiences in response to the spectrum. It would therefore be more accurate to call this the "inverted color argument." And because Locke's essential point is not limited to literal inversion, but could apply equally to any possible mapping of color experiences in one observer (e.g., you) to the same set of color experiences in another observer (e.g., me), we will call it the "transformed color argument." The problem it poses is whether any such color-to-color transformation (excluding the identity mapping) could accurately describe the relation between our color experiences without the differences being objectively detectable. Because we do not have direct access to each other's internal experiences, the question boils down to whether any such color-to-color transformation could exist without it being detectable through systematic differences in our behavior.

Notice that the transformed color argument as formulated above presupposes two important conditions: (1) that the two observers (canonically referred to as "you" and "me") have experiences in response to different spectra of electromagnetic energy and (2) that their overall sets of color experiences are the same. Dropping one or both of these assumptions leads to different versions of the more general "color problem." Relaxing (2) suggests the possibility that you and I both have experiences in response to different light spectra, but that one or both of us has at least some experiences the other does not. Our experiences might overlap to some degree, or they might even be completely disjoint, such that all of my chromatic experiences in response to light spectra are qualitatively different from any of yours. Relaxing (1) suggests that I might have no experiences of color whatsoever in response to different light spectra, yet behave as you do with respect to them. I would be a "color zombie," able to name colors properly and produce standard data in various color discrimination and matching experiments, but without having any corresponding experiences at all. These possibilities will come to the fore later in this article, but for now we will concentrate on the standard form of the problem in which it is assumed that you and I both have the same set of color experiences, whatever those might be, and ask whether they can be shown to be "differently arranged," so to speak.

The first problem is how we can get a scientific handle on this philosophical problem of whether transformed color experiences could be detected in publicly observable behavior. Locke presumed that we could not, but there are many scientifically documented aspects of color-related behavior that bear critically on the answer. I will argue that whether there exist any undetectable color-to-color transformations can be recast into the simple question of whether an empirically accurate model of human color experience contains any symmetries.

In scientific discussions, color experience is usually described in terms of spatial models. Perhaps the simplest and best known is the color circle devised by Newton (1704), but now familiar to artists and much of the general public (see Figure 1). It is an example of a color space: a multidimensional spatial representation (or model) in which different color experiences correspond to different points in the model. The locations of points representing colors are chosen so that the degree of psychological similarity between pairs of colors corresponds to the distance between the corresponding points in the model. In the color circle, the spectral colors of the rainbow are positioned in order along most of the circle's perimeter, and the nonspectral colors, including many reds, all magentas, and most purples, are located along the perimeter between the blue-violet and orange-red limits of the visible spectrum, as indicated in Figure 1 by the location of the color names outside the circle.

Figure 1

The color circle is a useful model of many aspects of color experience because it is easy to apprehend, yet manages to capture an immense number of facts about the relations among color experiences in a highly economical fashion. The fact that red is perceived as more similar to orange than it is to green, for example, is reflected in the fact that the point representing red is closer to the point representing orange than it is to the point representing green. Corresponding similarity relations among triples of color experiences -- that blue is more similar to purple than it is to yellow, etc. -- are thus faithfully preserved in the distance relations among triples of points in the color circle. This correspondence between psychological similarity relations and spatial proximity relations lies at the heart of Shepard's (1962a, 1962b) elegant nonmetric method of multidimensional scaling, and recovering the color circle from similarity data was one of his first demonstrations of its use.

We will now employ Newton's (admittedly simplistic) color space to illustrate how such models relate to the inverted spectrum argument and why Locke's intuition seems initially so compelling. Consider once again the case of literal spectral inversion, which we will henceforth call "rainbow reversal" to be perfectly clear. The hypothesis that my color experiences are rainbow-reversed relative to yours would mean that my color circle is the reflection of yours about the dashed line shown in Figure 1. The abbreviations inside the circle indicate the color experiences I have in response to the same lights that you experience as indicated on the outside of the circle. Thus, your red corresponds to my violet (and vice versa), your orange to my blue (and vice versa), your yellow to my cyan (and vice versa), and your chartreuse to my green (and vice versa), as indicated by the dashed arrows perpendicular to the axis of reflection. The question is whether this color-to-color transformation could be detected by behavioral means.

It might seem at first blush that similarity judgements among colors would reveal rainbow reversal, but, in fact, they would not. You would say that orange is more similar to red than to green, but so would I, even though this would correspond internally to my experiencing blue as more similar to violet than to chartreuse. Indeed, we would make all the same relative similarity judgements about rainbow colors despite the enormous differences in our internal experiences of them. With respect to color similarity judgments among rainbow experiences, then, Locke was right: Rainbow reversal cannot be detected behaviorally from such data.

The reason rainbow reversal is not behaviorally detectable from such color similarity judgements is that the empirical model they specify (i.e, the basic color circle) is symmetric about the axis of reflection that corresponds to reversing the rainbow. A symmetry in a spatial model is a transformation that maps the model onto itself such that it is the same before and after the transformation. Rainbow reversal is thus a symmetry of the color circle -- but so is any central reflection or rotation. Indeed, the only thing that makes any pair of such reflected or rotated color circles different is the nature of the internal experiences themselves. Because these are private events, any differences between yours and mine can only be assessed indirectly through our publicly observable behavior, as Wittgenstein (1953) argued so forcefully. The general claim is that any symmetry in a behaviorally constrained color space necessarily specifies a color-to-color transformation that cannot be detected by the behaviors that constrain the spatial model.

There is much more that we know about color experience from behavioral observations, however, and these facts must also constrain the color space whose symmetries we seek to understand. One important factor is the composition relations among colors: how some color experiences can be analyzed into combinations of other, more basic color experiences. Most colors of the rainbow are binary in the sense that they appear to be composed of (or analyzable into) two of the four primary chromatic colors: red, green, blue, and yellow.{1} Oranges, for example, seem to contain both redness and yellowness, purples to contain both blueness and redness, and so forth. In contrast, there are particular shades of red, green, blue, and yellow that appear not to be composed of any other colors.{2} For example, there is a particular red, called unique red, that appears purely red, with no traces of yellowness, blueness, or greenness in it. There are similarly defined unique colors for green, blue, and yellow, each of which is pure in the same sense of having no traces of the other primary colors. Moreover, pairs of these primaries are related to each other as polar opposites: red versus green, and blue versus yellow. These important hypotheses about compositional relations among color experiences were pointed out by Ewald Hering (1878/1964), who used them as the basis of his opponent process theory of color vision.

Relations of color composition are not the same as or reducible to the relations of color similarity discussed above. As the color circle shows, the similarity relations among red, blue, and purple are essentially the same as those among orange, red, and purple (see Figure 1). Even so, purple looks like it is composed of red and blue, whereas red does not look like it is composed of orange and purple. A complete model of color experience therefore requires that the uniqueness of the four chromatic primaries be represented within color space. It also requires a representation of the composition of colors in terms of how they are perceived to be analyzed into the four primaries.

Unique red, green, blue, and yellow can be represented by singular points at diametrically opposed locations in the color circle, and color composition relations of binary colors by their projections onto its orthogonal red-green and blue-yellow axes. Notice that this elaboration of the color circle now breaks many of the symmetries that were previously candidates for undetectable color-to-color transformations based only on color similarity relations. Only seven remain: the four central reflections about the dimensional axes and their angular bisectors, and the three central rotations of 90°, 180°, and 270°. Rainbow reflection has been eliminated because, if my color experiences were related to yours by this transformation, I would judge purple, chartreuse, orange, and cyan to be unique colors rather than red, green, blue, and yellow. All other transformations that map unique colors to binary colors are likewise eliminated.

It is now clear how one might proceed in a scientific evaluation of the transformed color argument: Document evidence of the existence of asymmetries in a behaviorally constrained model of human color experience. If all symmetries can be shown to be broken, then Locke was wrong: my internal color experiences cannot be a transformation of yours without the difference being detectable in my behavior relative to yours. The focus must be on finding asymmetries rather than finding symmetries because the nature of scientific hypothesis testing requires ruling out null hypotheses (i.e., finding that differences are present) rather than accepting them (i.e., failing to find such differences). In the context of symmetries, this amounts to detecting asymmetries where there are differences rather than detecting symmetries where there are no differences.

1.3 ASYMMETRIES IN LIGHTNESS

The color circle we have been using as a model of human color experience is quite inadequate, however, primarily because it leaves out the vast majority of color experiences, including white, black, all their mixtures with each other (grays), and all their mixtures with the chromatic colors along the perimeter of the color circle. These color experiences are missing because the color circle is only two-dimensional, whereas the full set of color experiences is three dimensional. Figure 2 shows a more complete spatial model of human color experiences that reflects color similarity relations in all three dimensions of perceived surface color and the full set of six of compositional primary colors: red, green, blue, and yellow (as before), plus black and white.{3}

Figure 2

The experiences of surface colors represented in Figure 2 are classically defined by three dimensions, which we will call hue, saturation, and lightness.{4} (For lights, the third dimension is usually designated as brightness rather than lightness.) The dimension we normally associate with the basic "color" of a surface or light is called its hue. In color space, hue corresponds to the angular direction in the horizontal plane from the central axis of color space to the location of the point representing that color. The second dimension of color space, called saturation, represents the "vividness" of color experiences. It corresponds to the perpendicular distance from the central axis to the position of the color experience in color space. For example, the vivid colors of the rainbow lie along the outside edge because they have maximum saturation. All the grays lie along the central axis because they have zero saturation. The "muted," "muddy," and "pastel" colors in between have intermediate levels of saturation. The third dimension of surface color is lightness. In the coordinates of color space, lightness refers to the "height" of a color's position as it is drawn in Figure 2. All surface colors have some value on the lightness dimension, although it is perhaps most obvious for the achromatic grays that lie along the central axis, with white at the top and black at the bottom. In particular, it is worth noting that browns are represented as dark yellows and oranges (i.e., ones low in lightness), as indicated in Figure 2.

The color circle corresponds to the perimeter of an oblique section through this color solid. This section is oblique because the most saturated yellows are quite light (and therefore higher in color space) whereas the most saturated blues and purples are quite dark (and therefore lower in color space), with the most saturated reds and greens at intermediate lightness values. This variation in lightness of maximum saturation colors breaks further symmetries in color space. A rotation of 180° about the vertical axis, for example, would map yellow to blue and blue to yellow. This color transformation could be detected behaviorally, however, because you would judge unique yellow to be lighter than unique blue, whereas I would judge the reverse. Because the color solid has no rotational symmetries, any simple rotation of color space about its lightness axis can be ruled out as a behaviorally undetectable color-to-color transformation. That is, rotations of 3-D color space can be detected and cannot be used to support the color transformation argument.

There are still three approximate reflectional symmetries of the 3-D color spindle shown in Figure 3 that are likely to escape behavioral detection except in the most precise psychophysical tasks. One is reflection of red and green in the blue-yellow-black-white plane (Figure 3A). This works because, at least to a first approximation, red and green are the same in lightness, and blue and yellow are mapped to themselves. A second symmetry is reversing both the blue-yellow and black-white axes (Figure 3B). This solves the lightness problem with reversing blue and yellow because it also reverses the lightness (black-white) dimension. The third symmetry is the composition of the other two: namely, reversing all three axes, red-for-green, blue-for-yellow and black-for-white (Figure 3C).

Figure 3

Although all three of these transformations are logically possible, by far the most plausible is reflecting just the red-green dimension. Indeed, a persuasive argument can be made that such red-green reversed perceivers may actually exist in the population of so-called "normal trichromats" (see Nida-Rümelin, 1996). The argument goes like this. Normal trichromats have three different pigments in their three cone types. Some people, called protanopes, are red-green color blind because they have a gene that causes their long-wavelength (L) cones to have the same pigment as their medium-wavelength (M) cones. Other people, called deuteranopes, have a different form of red-green color blindness because they have a different gene that causes their M-cones to have the same pigment as their L-cones. In both cases, people with these genetic defects lose the ability to experience the red-green dimension of color space because the visual system codes this dimension by taking the difference between the outputs of the M- and L-cones. Now suppose that someone had the genes for both forms of red-green color blindness simultaneously. Their L-cones would have the M-pigment, and their M-cones would have the L-pigment. Such people, would therefore not be red-green color blind at all, but simply red-green reversed trichromats.{5} They should exist. Assuming they do, they are proof that this color transformation is either undetectable or very difficult to detect by purely behavioral means, because nobody has ever managed to identify one.

Harrison (1973) and Hardin (1997) have argued that there are further hurdles for transformed color arguments to clear concerning the implications of color categories. Although it is not entirely clear that such categories reflect basic facts about color experience rather than some later cognitive stage of processing, if they do, they have important consequences for the behavioral detectability of color transformations. To explain their import on color transformation arguments, we will have to review briefly some background claims about color naming and categorization.{6}

In their ground breaking studies of cross-cultural color naming, Berlin and Kay (1969) found that there are a very small number of basic color terms (BCTs) across all the languages that they studied. These BCTs refer to a corresponding set of basic color categories (BCCs) into which color experience can be partitioned. (We will assume the obvious one-to-one mapping between BCTs and BCCs in the discussion that follows.) Further research and analysis has postulated three different types of BCTs: primary, derived, and composite (Kay & McDaniel, 1978). The most basic are the six primary categories: RED, GREEN, BLUE, YELLOW, BLACK, and WHITE. From these, six more categories are "derived" by the fuzzy-logical AND-ing (via fuzzy-set intersection; see Zadeh, 1975) of two primary color categories:

GRAY = WHITE AND BLACK,

ORANGE = RED AND YELLOW,

PURPLE = RED AND BLUE,

BROWN = BLACK AND YELLOW,

PINK = WHITE AND RED,

GOLUBOI (a Russian word) = WHITE AND BLUE.{7}

Notice that this set does not include all possible combinations of primary BCCs. Some are ruled out by the structure of color space itself, such as red-green and blue-yellow, which cannot exist because they simply do not overlap and therefore have no exemplars in the their fuzzy-logical intersection. Other combinations could exist as BCTs, such as blue-green, but do not for as-yet-unknown reasons.

The four "composite" color categories are formed by the fuzzy-logical OR-ing of two or more primary color categories:

WARM = RED OR YELLOW,

COOL = GREEN OR BLUE,

LIGHT-WARM = WHITE OR WARM = WHITE OR RED OR YELLOW,

DARK-COOL = BLACK OR COOL = BLACK OR GREEN OR BLUE.

Again, not all possible combinations of primary BCCs exist as composite BCTs. It seems reasonable that they be restricted to combinations of nearby primary BCCs in color space, ruling out RED OR GREEN and BLUE OR YELLOW. But it is not clear why there are either no or few composite BCTs in known languages for RED OR BLUE, GREEN OR YELLOW, WHITE OR COOL, or BLACK OR WARM. These and other mysteries remain to be solved.

It is important to realize that these facts about BCTs are relevant to the present discussion only if they reveal important asymmetries in the structure of human color experiences. For example, if for some reason there are more just-noticeable-differences (jnds) between unique red and unique yellow than there are between unique green and unique blue, the wider psychophysical gap might explain why there are BCTs for ORANGE in many languages, but not for CYAN (blue-green). The fact that there are strong (possibly even universal) constraints on the BCTs that have been discovered in a large number of natural languages suggests that some basic neural mechanisms of human color vision are likely to be responsible. The most plausible alternative explanations are that the constraints on BCTs reflect structure in the nature of the environment (e.g., perhaps there are more salient orange-colored objects than cyan-colored objects), the nature of an organism's fit to its ecological niche (e.g., perhaps distinctions between different shades around orange are more important to the organism than those around cyan), or the patterns of contact and influence between languages.

If the empirical constraints on BCTs do, in fact, reflect underlying inhomogeneities in the structure of color experience, they can be used to break further symmetries of color space. Primary BCTs, for which the evidence of universality is strongest, actually do not break any of the symmetries illustrated in Figure 3. This is because they correspond to the six Hering primaries (red, green, blue, yellow, black, and white) described above, which have already been taken into account by the unique points and axes of the color space.

Adding the derived and composite BCTs, however, breaks all further symmetries. Consider first derived BCTs. Red-green reversal (Figure 3A) is ruled out, for example, by the asymmetry between the frequency of derived BCTs for ORANGE and PURPLE (frequently found) versus CYAN and CHARTREUSE (infrequently or never found). If I were red-green reversed -- and if derived BCTs reflect intrinsic inhomogeneities in my color experience -- I should find it strange that BCTs were distributed in this way rather than in the opposite way. I should also find it odd that there is a BCT for PINK rather than light-green. These facts pose no problem for blue-yellow and black-white reversal (Figure 3B), however, because ORANGE maps to PURPLE (and vice versa), and PINK maps to itself.

Table 1 shows how different BCTs break the three candidate symmetries illustrated in Figure 3. The entries in this table were generated from Kay and McDaniel's (1978) analysis as follows. First, the transformation indicated at the top of each column dictates the remapping of primary BCCs as shown in the first six rows. Red-green reversal (column 1), for example, only requires changing RED to GREEN' and GREEN to RED, where the first term indicates the original system of BCCs and the second designates the transformed system of BCCs. Next, the appropriate substitutions are made in Kay and McDaniel's formulas (see above) for the derived and composite BCCs. If the resulting formula for a new BCC corresponds to one of an old BCC, then a "+" is placed in the column for that transformation and the row of the original BCT. If not, then an "X" is entered there. To illustrate, consider the row for PURPLE. After red-green reversal (column 1 of Table 1), PURPLE = RED AND BLUE becomes PURPLE = GREEN AND BLUE. Because GREEN AND BLUE is not the formula for a BCT in Kay and McDaniel's theory, an "X" is indicated in the PURPLE row of the first column, meaning that this transformation maps this BCC into a nonBCC. After blue-yellow and black-white reversal (column 2 of Table 1), however, PURPLE = RED AND BLUE becomes PURPLE = RED AND YELLOW. Because RED AND YELLOW is the formula for a BCT (namely, ORANGE), a "+" is entered in the table, indicating that this transformation maps a BCC into another BCC. For complete reversal (column 3), PURPLE = RED AND BLUE becomes PURPLE = GREEN AND YELLOW, which is not a BCT; hence the "X" in column 3.

The validity of this analysis rests on the validity of Kay and Mc Daniel's original theory, of course, including the set of BCTs they enumerate and their definitions in terms of fuzzy logical operations. If new BCTs have been discovered in the meantime or if new formulas have been proposed, the analysis of Table 1 will be correspondingly wrong in detail. But the general nature of the reasoning is sound within this qualitative theoretical framework, and a more complete or accurate theory can be substituted for Kay and Mc Daniel's original one. Notice also that the analysis in Table 1 does not involve metric evaluations of asymmetries, but only approximate, qualitative ones. As long as there is a BCT in the general neighborhood of the transformed BCT, the relation is counted as symmetrical (+) in the table. As it turns out, metric precision is unnecessary because the symmetries are broken qualitatively.

The analysis in Table 1 shows that no color-to-color transformations survive a thorough-going BCT analysis intact, indicating that all symmetries are broken by the behavioral constraints implied by BCTs. I am not aware, however, of any behavioral data that directly support these asymmetries for derived and composite color categories in color experience. In many cases, the difference between derived or composite BCTs and nonBCTs is subtle enough that direct introspections are too blunt an instrument to decide. I myself would be hard-pressed to claim, for example, that it seems "better" or "more natural" to me that there is a BCT for light-reds (PINK) than for light-greens, independent of the fact that my language actually has a BCT for light-reds and not for light-greens. The case for ORANGE and PURPLE over blue-green and yellow-green seems somewhat more compelling. Even so, it would be hard to tell how much of such preferences for derived and composite BCCs over nonBCCs is the product of sociolinguistic training rather than asymmetries in my underlying color experiences.

The existing evidence most relevant to these asymmetries comes from Rosch/Heider's studies of learning color terms in the Dani tribe of New Guinea (Heider, 1972). In a classic cross-cultural experiment, Rosch found that the Dani, who have BCTs only for LIGHT-WARM and DARK-COOL, were able to learn new categories for RED, BLUE, GREEN, and YELLOW more easily than new categories for ORANGE, PURPLE, CYAN, and CHARTREUSE. This result shows that primary BCCs appear to be preferred over other color categories for the Dani -- and presumably other cultures with composite BCTs -- even though these categories are not overtly expressed in the BCTs of their language.

It is not yet clear whether this distinction would also be supported for derived or composite BCTs -- which are the only ones that break the symmetries in Figure 3 -- because Rosch's studies with the Dani examined the learnability only of primary BCTs. Some derived BCTs were used in the study (ORANGE and PURPLE), but they were actually employed in the contrasting nonBCT "control" categories. Moreover, her results, which have traditionally been interpreted in terms of the learnability of primary BCTs, can equally well be explained by color composition relations based on the four chromatic Hering primaries. The latter explanation has the advantage of a clearer basis in phenomenology and physiology than is available for BCTs in general.

The main question is whether the derived and composite BCTs are firmly grounded enough in color experience for the asymmetries they imply to be detected. Perhaps a new category for light-green would be just as easy to learn as PINK for people whose language has neither BCT, and perhaps a new BCT for blue-green or yellow-green would be just as easy to learn as ORANGE or PURPLE. The strongest argument for a phenomenologically privileged status of a derived BCC can be made for BROWN, because it seems qualitatively different from the yellow and orange hue families of which it is part (Hardin, 1997).

The crucial question at the center of this issue is whether the structure associated with BCCs is due to inhomogeneities of color experience. The most obvious way to document such effects of categories on perceptual experience is to look for so-called "categorical perception" phenomena. Categorical perception refers to a phenomenon in which small changes in certain stimulus continua across a categorical boundary produce large changes in perceptual experience, whereas corresponding changes within category boundaries produce much smaller changes in experience. (See Harnad, 1987, for a general discussion of this topic.) The classic case is the effect of continuous acoustical variables on categorical perception of phonemes (e.g., Liberman, Harris, Hoffman & Griffith, 1957).

In the color domain, the evidence is mixed. On the one hand, categorical effects in color perception have been reported by several researchers, even in infants (e.g., Bornstein, Kessen & Weiskopf, 1976) and monkeys (e.g., Sandell, Gross & Bornstein, 1979) for whom linguistic labels cannot be the mediators of such effects. On the other hand, these categorical effects are seldom as sharply defined as for categories of speech perception, and the fuzziness and ineffability of category boundaries is well documented in many other studies (e.g., Berlin & Kay, 1969; Rosch, 1973). Moreover, what categorical effects have been reported are typically restricted to the primary BCTs of red, green, blue, and yellow, leaving us, once again, with an open question about the status of derived BCTs.

One phenomenon of normal experience that can be viewed as supporting categorical structure in color experience is the banded appearance of the rainbow Hardin (1997). The physical continuum of photon wavelength that underlies the rainbow is purely quantitative and unidimensional, with no physical divisions that would produce "bands" of any sort. Why, then, does a wavelength rainbow appear banded? One possibility is that qualitative distinctions between color categories are directly represented in perceptual experience, as Hardin (1997) has argued, and that these produce qualitatively distinct bands in the appearance of the rainbow.

There is an alternative to the categorical explanation that must be considered, however. Because all chromatic colors (except the four unique ones) are experienced as mixtures of different amounts of red, green, blue, and yellow, the banded appearance of the rainbow might arise simply from the gradual transitions between these qualitatively different colors. In this case, the bands are attributable to color composition rather than color categories. The most obvious question to discriminate between these two possibilities is whether orange is perceived as a distinct, qualitatively different band from the adjacent reds and yellows, whether it is perceived merely as a transition between them, or whether it is something in between. The BCC view predicts a separate orange band because of the existence of the derived color category for orange, whereas the compositional view predicts no such band. If people do experience a separate orange band, there is the further question of whether this band is present only in the perceptions of people who speak languages with a BCT for ORANGE or whether it appears universally. Unfortunately, we do not yet have the answers to these deceptively simple questions.

Whether derived and composite BCTs are grounded in color experience may seem like a fine point, but, as Table 1 shows, it has crucial implications for the possibility of behaviorally detecting color transformations. If BCCs are not reflected in color experience, or if only primary BCCs are, then the prior conclusion stands that there are at least three transformations of color space that may well escape behavioral detection. If composite and/or derived BCCs are relevant, then no form of the color transformation argument will actually work.

Thus far, we have been assuming that all aspects of color experience can be naturally represented within a spatial model like the one shown in Figure 2, but this is not necessarily true. One potential problem concerns systematic asymmetries in color similarity relations. An axiomatic property of all metric dimensional spaces is that distances between points are symmetrical: the distance from A to B is the same as that from B to A (Krantz, Luce, Suppes, & Tversky, 1971). If spatial distance is to represent experienced (dis)similarity, then color similarity relations must also be symmetrical.

Rosch (1975) has reported similarity results that contradict this assumption with respect to focal versus nonfocal colors (which correspond approximately to unique and binary colors, respectively). When Rosch had subjects indicate the perceived similarity between one color (the target) and another (the standard), she found small but systematic effects: Nonfocal targets were perceived as more similar to focal standards than vice versa (e.g., off-reds were judged more similar to true-red than true-red was to off-reds). Although these effects were not large, they are noteworthy for at least two reasons. One is that they create a serious problem for capturing all relations among colors in a purely spatial model. The other is that they may constitute another kind of evidence that color categories influence color experience.

Even so, Rosch's results do not necessarily rule out the possibility that certain color transformations can escape behavioral detection. There are two issues. The first concerns how these asymmetries in similarity are distributed in color space. The focal colors for the primary BCCs are essentially the unique primaries, and we have already noted that the three transformations of Figure 3 preserve their uniqueness. Asymmetrical distortions in distance relations with respect to these unique points can also be preserved by certain transformations, as illustrated in Figure 4. This diagram shows a hypothetical representation of asymmetries in similarity relations within the color circle that would be preserved by the same reflections that preserve the four unique points. The magnitude and direction of the asymmetries are represented by vectors indicating the directional difference in similarity between the four primary focal colors (large circles) and various comparison colors (small circles with arrows). This diagram shows that such asymmetries in similarity could be symmetrically distributed in a color space with respect to the primary focal colors. We simply do not have enough information on this issue to arrive at a firm conclusion.

Figure 4

The second critical issue for the behavioral detectability of color transformations is whether asymmetrical similarities to focal colors hold just for primary categories or whether they also hold for derived categories. As we have already noted, derived categories break all global symmetries of color space, so asymmetries in similarity with respect to these focal colors (e.g., ORANGE, PURPLE, PINK, BROWN, and GOLUBOI) would allow detection of any color-to-color transformations. On this point, I know of no evidence. Rosch found asymmetries in color similarity using only the primary focal colors -- RED, GREEN, BLUE, and YELLOW -- but did not test for asymmetries in derived color categories. The primary focal colors are symmetrically distributed and thus may cause no problems for any of the candidate symmetries of color space in Figure 3. A further question is whether these asymmetries are actually due to color categories or to color composition, which has a clearer and more obvious bearing on color experience. Again, we do not yet know the answer.

There are other potential sources of asymmetry in color space that might reflect inhomogeneities in color experience that could be detected behaviorally. One concerns the metrical structure of color space as measured by the discriminability of color samples. Suppose, for example, that unique red is perceived to be more different from unique blue than unique blue is from unique green. Given that they are all unique versions of chromatic primaries, it seems plausible that they are, in some sense, equally different, and this is the rationale for placing them at opposite poles of orthogonal diameters of the color circle in Figure 1. But there are other ways of determining distances in color space psychophysically, such as counting the number of jnds between pairs of colors. Each jnd is measured psychophysically by finding the difference threshold: the smallest difference along a continuum that can just barely be detected. Using this method, red and blue might well prove to be more different from each other than blue and green. Munsell color space, which is based on measurements of equally-spaced differences in hue, represents this difference in discriminability by a greater distance between red and blue, and MacLaury (1987) has reported data supporting the same conclusion. Other metrical differences might also prove to be asymmetric when measured by counting jnds, and if they are reliably different, they break what otherwise might be plausible symmetries.

Let us summarize our discussion about the empirical possibility of detecting color transformations empirically. There are just three candidate transformations that survive the most basic behavioral constraints concerning color experience as reflected in the global structure of color space: red-green reversal, blue-yellow and black-white reversal, and reversal of all three axes (red-green, blue-yellow, and black-white), as illustrated in Figure 3. If the color space depicted in Figure 2 is at least roughly accurate, then, all three transformations will preserve the similarity relations among colors, the dimensional description of colors in terms of hue, saturation, and lightness, the decomposition of colors into the six Hering primaries, and the distinguished points of the six unique colors.

Adding color categories and color naming data into the mix makes the situation more complex. The same three transformations survive further constraints due to the primary BCCs and BCTs, including the distribution of the six primary color categories in color space and the asymmetries in similarity relations around the focal colors for the four chromatic primary color categories. Composite and derived BCCs, however, rule out all transformations by breaking their symmetries, but only if they are due to intrinsic properties of the color system (i.e., based on experiential factors) rather than to extrinsic ecological factors (i.e., based on the physical environment or sociolinguistic community).

I have argued that the crucial issue in assessing the validity of the transformed color argument is the existence of symmetries in an empirically accurate model of color experiences. Because no such model presently exists, the exercise is premature for reaching firm conclusions. I have used color spaces as the focus of this enterprise because they are by far the dominant modeling tool for this domain and because the nature of global transformations are particularly transparent within them. The argument from symmetry is not limited to spatial models, however, for it can be applied to neural network models, abstract propositional models, or any other sort of model. The only requirements are that the set of possible color-to-color transformations can be specified in the model and that the results of such transformations can be assessed in terms of the requisite empirical constraints. If the behavior of the model is invariant over the transformation, it is symmetric with respect to that transformation, and the transformed color argument will work.

2 THE ISOMORPHISM CONSTRAINT

The questions to which I now turn concern which aspects of mental life scientists can hope to study and understand objectively and which we cannot, using color experiences as the example. In the present section I will discuss the limitations of behavioral science, and in Section 3, I will consider the possibility that biological science can take us beyond these limits.

It is universally agreed that there is a behaviorally defined subjectivity barrier with respect to how much others can know about our experiences, and color experiences are no exception. Some aspects of experience are shared across observers, whereas others cannot be. We know that many aspects of color experience must be shared across observers because normal trichromats agree in our linguistic statements and other sorts of discriminative behavior with respect to colors. Color blind individuals also agree with others having the same form of color deficiency, but they do not agree across color-deficiency classes or with normal trichromats. These aspects of color experience are therefore objectively shared and fully available to behavioral science. Other aspects are indeterminate in this respect, however, in that they appear to be free to vary without affecting any known aspect of behavior. They are purely subjective and therefore unavailable to behavioral science. Even if they happen to be identical across observers, scientists would never know with certainty that this was the case. In this section I attempt to define this barrier between objective and subjective phenomena with respect to behavioral science.

I claim that the two relevant aspects of experience are the intrinsic qualities of experiences themselves versus the relational structure that holds among those experiences. These two aspects are normally so completely intertwined that it may seem perverse to advocate separating them, but if they lie on different sides of the subjectivity barrier, as I claim, then it is important to make the distinction.

Certainly the most obvious aspect of visual awareness is the nature of the experiences themselves, such as the sensory quality of redness or circularity, to pick two examples at random. It seems that the quality of these experiences are flat-out impossible to define behaviorally, given that nobody has access to anyone's experiences except his or her own. This is why color-to-color transformations are a legitimate problem to begin with: The quality of individual experiences lies beyond the behavioral subjectivity barrier.

One might suppose that there is at least one aspect of experiences that can be specified behaviorally: namely, their individuality. The set of colors that a person can individuate (discriminate), for example, determines whether someone has full trichromatic color experience or a restricted set due to some form of color blindness. But notice that experiences can be individuated behaviorally only by asking people to discriminate between two stimuli, responding "same" or "different" to various pairs. Colorblind individuals reveal their reduced set of color experiences by performing at chance in discriminating between certain color samples that normal individuals distinguish quite easily. Thus, even individuating experiences behaviorally is actually about the relation between two (or more) experiences by designating whether they are the same or different.

The second aspect of experience is one to which behavioral science does have access: namely, structure among experiences carried by their relations to each other. Regardless of what the experience of red itself is like, normal trichromats agree that it is more like orange than green. Likewise, sighted observers agree that a circle looks more like a regular octagon than an equilateral triangle. These relations among experiences are just as important -- and in certain respects, even more so -- than the qualities of experiences per se because they determine the structure of experience, which can be shared despite the subjectivity barrier. If experiences had no relational structure, they would simply be a collection of completely different and totally unrelated mental states, like the "blooming buzzing confusion" which William James suggested to be the nature of sensory experience in infants (James, 1890/1950). Without relational structure, for example, we would not experience colors as being a coherent domain of experience, more similar to each other than they are to shapes: Redness would be as much like circularity as it is like greenness -- or middle-C, or the smell of freshly ground coffee, or the taste of pumpkin pie.

Relational structure is even more crucial within a single domain of experience such as color. Without it we would not experience white as being lighter than gray or gray as being lighter than black; we would only experience them as different colors, and equally different at that. Because of lighter-than relations among color experiences, we are aware of the ordering of colors in terms of the continuous dimension of lightness, ranging from black to white. Indeed, the entire structure of color space (see Figure 2) is determined by relations among colors, particularly relations of composition and similarity. They provide the rich, complex dimensional superstructure of color experience. It is quite literally unimaginable what color experience would be like without this structure.

I have argued at some length that it may not be possible to be sure that my experiences of colors are the same as your experiences of colors strictly from our behavior, since mine could just as easily be some structure-preserving transformation of yours. We can (and do) agree on basic color terms that refer to them -- I call roses "red", violets "blue", and so forth -- so that we can communicate effectively about individual colors. This is an objective behavioral fact about color experience, but it tells us absolutely nothing about the quality of those experiences except that they are discriminably different from others. It is an exceedingly weak constraint in the same sense that a nominal scale is the weakest type of measurement system (Stevens, 1951). We just learn to attach the same labels to our corresponding internal experiences that arise from viewing the same collections of wavelengths, regardless of what our particular internal experiences might be.

Further constraints are introduced, however, once we begin to consider binary or higher-order relations among experiences (Krantz et al., 1971). Both you and I can make judgements about the relative similarity of two colors to a third, or the relative lightness of two colors, for example. These inherently relational judgements are also objective in that normal trichromats agree about them, at least within some margin of error. This is not to say that my relational experiences are the same as yours or that we can even determine whether they are or not. My experiences of color similarity relations might be as wildly different from yours as my individual color experiences are from yours. But the structure of our experiences and relations can nevertheless be identical.

Preserving relational structure appears to be a necessary condition for one set of objects to represent another (Palmer, 1978). Indeed, model theory formalizes the situation in which one set of objects models another in terms of there existing a function that maps objects in the first set to objects in the second set such that corresponding relations are preserved in a precise, set-theoretic sense (Tarski, 1954).{8} This requirement explains why the same 3-D color solid is able to model color experiences for all normal trichromats, even if they happen to have wildly different color experiences: It captures exactly the relational structure among the color experiences for each individual by mapping them onto points in space such that relations among color experiences are preserved by spatial relations among points in the color solid (see Figure 5). This is not to say that my experience of white is literally above my experience of black, even though the point representing white in the color solid is above the one representing black in the conventional color solid. But my experience of white is lighter-than that of black, and the relation above in canonical color space corresponds to the relation lighter-than in color experience and preserves its structure.

The emerging picture is that the nature of color experiences cannot be uniquely fixed by objective behavioral means, but their structural interrelations can be. This means that, logically speaking, any set of underlying experiences will do for color, provided they relate to each other in the required way. The same argument may be extendable quite generally to other perceptual and conceptual domains, although both the underlying experiential components and their relational structure will be different. The experience of musical pitch, for example, could be grounded in any of an infinite variety of experiential dimensions, but it would always have to have the same double helical structure characteristic of perceived pitch relations (Shepard, 1982). Although both of these examples are cases in which there is a clear geometrical structure associated with the experiential domain, this need not be true. The only requirment is that there be some kind of relations among the experiences that constitute their structure.

In Section 1 I argued at some length that the existence of symmetries in color space is the key issue in assessing the validity of color transformation arguments. I now return to this topic to ask why this might be the case and how the answer relates to the foregoing discussion of the structural of experiences.

Mathematically, symmetries are functions that have two special properties known as automorphism and isomorphism. They are automorphisms because they map a given domain onto itself in a one-to-one fashion. In the case of symmetries in color space, the automorphic function therefore maps a color space onto itself. Automorphism is critical for Locke's argument because, as was mentioned at the outset, it assumes that the two observers in question both have the same overall set of color experiences. The only question is whether their color experiences might have different causal connections to the outside world.

The second special property of symmetries is that they are isomorphisms. Isomorphism, which means roughly "having the same structure", is a mathematical function in which, intuitively speaking, one set of entities is mapped onto another set of entities such that the structure of relations among the first set are preserved by the structure of corresponding relations among the second set (e.g., Tarski, 1954). {9} Figure 5 illustrates the general requirements for an isomorphism to hold using the (presumed) isomorphic relation between color experience and color space as an example. The function maps color experiences onto points in a dimensional color space such that relations among color experiences (lighter than, more similar than, etc.) are preserved by corresponding relations among corresponding points in space (higher than, closer to, etc.). This allows a direct and valid translation from facts about the relations among color experiences into facts about the relations among points in color space. The only thing that is missing is the capability of saying anything about the nature of the color experiences themselves (or the nature of the points themselves) except that they satisfy corresponding relations. The fact that such an isomorphism can be constructed is the principle reason that spatial models are good representations of color experiences. Notice further that because isomorphism is both transitive and symmetric, if your color experience and my color experience are isomorphic to the same color space, then our color experiences are necessarily isomorphic to each other.{10}

Figure 5

Isomorphism is crucial for the transformed color argument because, I claim, the only kinds of differences that can be detected behaviorally are differences in relational structure, and relational structure is precisely what is preserved by isomorphism. I can say (or otherwise indicate behaviorally) that color A is lighter than color B, but cannot in any way communicate how light either one appears to me in absolute terms. I can also say with confidence that red is more similar to orange than it is to green, but cannot express either similarity in absolute terms. It might seem that one can make "absolute" ratings of, say, the lightness of individual color samples on a rating scale, but such ratings are, in fact, always relative to the range of possibilities.

Both automorphism and isomorphism are required to satisfy the assumptions of Locke's original inverted spectrum argument. There are other versions of the more general color question that do not require the automorphism component, however. If you and I both have color experiences, for example, but they are not the same overall set, then automorphism does not hold. The nonoverlap can vary from minimal to complete. It would be minimal, for example, if everything seemed just a bit lighter to me than to you. In this case, no new dimensions are involved, just different values over an expanded range of lightnesses. My experience of pure white would then be lighter than any experience you have ever had, and your experience of pure black would be darker than any experience I have ever had. This mapping between our experiences is not automorphic because there are some color experiences -- my white and your black -- are unique to one of us.

More radically, though, my color experiences might be totally different from your color experiences in ways neither you nor I can imagine. This would be the case if my experiences of the three dimensions of color space were qualitatively different from yours, as though we lived in completely different subspaces of some hypothetical "experiential hyperspace." I would have chromatic dimensions for what we all call red-green and blue-yellow variations, just as you would, but they would span hue dimensions qualitatively different from any you have ever experienced. The existence of such additional experiential dimensions of color can be inferred from comparative studies of color vision, which show that some animals have four or even five dimensions of color experience (see Thompson, Palacios & Varela, 1992). At least some of the dimensions of chromatic experiences of such animals, whatever they may be, must be qualitatively different from any of yours and mine. There is certainly no logical requirement that my experiences of the range of hues (or saturations or lightnesses) be anything like your experience of them. Biological considerations can be brought to bear, as we will discuss shortly at some length, but there are enough differences between the brains of different perceivers to undermine an a priori assumption that the color experiences they give rise to are necessarily the same, except in extraordinary circumstances such as exact clones.

Such considerations lead me to conclude that automorphism is not central to understanding the general question of behavioral detectability of differences in color experiences, even though it is required for Locke's inverted spectrum argument. Isomorphism, however, appears to be key in evaluating the detectability of any color transformations under any circumstances. If your color experiences are isomorphic to mine, your experiences will be undetectably different from mine because the structure in the relations among your color experiences is the same as that in the corresponding relations among my color experiences. And if only relations can be assessed by behavioral means, then isomorphism is the strongest form of equality that can be claimed for color experiences across observers based on behavior. The relations that must be structurally preserved include (at least) color similarity, color composition, unique vs. binary colors, and dimensional ordering.

I will call this condition of structural equality to the level of isomorphism the isomorphism constraint and claim that it constitutes a fundamental limitation on what can be discovered about experience by behavioral methods. It means that, even if all the dimensions of my color experiences are qualitatively different from yours, we can still behave identically with respect to colors as long as our experiences are isomorphic.{11} My experiences would clearly have to be three-dimensional, would have to include six unique reference experiences (for the unique colors) at the poles of three axes, would have to include an angular dimension for hue, a radial dimension for saturation, and a linear dimension for lightness, and so forth. If all the relevant conditions were met, then my color experiences could be arbitrarily different from yours without the differences being behaviorally detectable.

Again, it is important to understand that none of these conclusions depends on there being spatial models of the cognitive domain. Experiential domains in which there are viable spatial models make good illustrations because the idea of isomorphism is particularly clear in such cases, but spatial models are by no means necessary. The internal relations among experiences could even be fundamentally incompatible with spatial representations, conforming to none of the fundamental axioms of metric dimensional spaces. The only requirement is that, whatever the qualitative nature of your internal experiences and the relations among them may be, the relations among my corresponding experiences must have the same structure.

If indeed the shared aspects of experience coincide exactly with structural relations -- i.e., that which is preserved by an isomorphism -- the argument thus far can be summarized as follows: objective behavioral methods can determine the nature of experiences up to, but not beyond, the criterion of isomorphism. The subjectivity barrier would then coincide precisely with the isomorphism constraint.

Interestingly, this criterion of isomorphism is not unique to the subjectivity barrier or to behavioral science, for it also exists in axiomatic formulations of mathematics. In classical mathematics, a domain is formalized by specifying a set of primitive elements (e.g., points, lines, planes, and 3-D spaces in geometry) and a set of axioms that specify the relations among them (e.g., two points uniquely determine a line, three noncollinear points a plane, etc.). Given a set of primitive elements, a set of axioms, and the rules of mathematical inference, mathematicians can prove theorems that specify many further relations among mathematical objects in the domain. These theorems are guaranteed to be true if the axioms are true.

But the elements to which all the axioms and theorems refer cannot be fixed in any way except by the nature of the relations among them; they refer equally to any entities that satisfy the set of axioms. That is why mathematicians sometimes discover that there is an alternative interpretation of the primitive elements, called a dual system, in which all the same statements hold. For example, the points lines, planes, and spaces of projective geometry in three dimensions can be reinterpreted as spaces, planes, lines, and points, respectively, because all the same relations hold when the elements in the latter system are substituted systematically for their corresponding dual elements in the former system. All the same axioms hold, and therefore all the same theorems are true. An axiomatic mathematical system can therefore be conceived as a complex structure of mathematical relations on an underlying, but otherwise undefined, set of primitives that are free to vary in any way. As Poincaré (1952) observed, "Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change." (pp. 20) The same can be said about behavioral scientists with respect to consciousness: we do not study experiences, but the relations among experiences. It is (or should be) a matter of indifference to behavioral scientists if the experiences of one person differ from those of another, provided that the relations among experiences are the same.

2.4 RELATION TO FUNCTIONALISM

The analysis we have just given of color experience bears important relations to certain aspects of functionalism. One salient characterization of functionalist accounts of the mind is that they are based on the causal relations among mental states and their input and output relations to the external world (Dennett, 1978; Fodor, 1968; Putnam, 1960). Two cognitive systems that have the same causal relational structure (i.e., that have corresponding causal relations among all their corresponding mental states) and the same causal relations to the external world (on both the input and the output ends) are functionally equivalent. Functionalist doctrine claims that such systems -- which we can call causally isomorphic (but not causally equivalent) -- are therefore mentally equivalent.

However, the analysis in Section 1 suggests that this is not necessarily so. In particular, two systems having the same causal relational structure (including input-output relations to the environment) may have radically different conscious states. There are at least two ways in which this can happen. One was discussed at length in Section 1: namely, the possibility that the experiences of two people may be quite different even though they have all the same relational structure, as in the behavioral undetectability of causally isomorphic color experiences. The other is that one of the systems might have the same causal relational structure over its internal states and their relations to the world, yet have no conscious experiences whatever. Let us now consider this latter case more carefully.

Suppose we were to create a working "color machine" that actually processes information in light in the same way as people do and that responds as people typically do. This is a reasonable goal. Figure 6 illustrates one way to construct the "front end" of such a machine. It analyzes incoming light using prisms, cardboard masks, photometers, electronic adding and subtracting circuits, and so forth to process color information according to the principles of color perception as they are currently understood. The details of how we now believe receptor outputs to be integrated to compute the dimensional values of color space may be wrong, of course, but the crucial issue is whether substituting the right computations would result in the machine having color experiences or not. To enable such a machine to respond behaviorally to light patches, it would have to be extended by adding processes to produce basic color terms for the colors it is shown, to analyze the composition of colors into their compositions in terms of the Hering primaries, to make color similarity ratings, and so forth. Moreover, it would have to do all this in a way that is behaviorally and computationally equivalent to the way in which people perform these tasks. Supposing that such a machine could be constructed -- and it would not be very difficult to do -- it seems almost bizarre to claim that because it derived the correct coordinates in color space for, say, unique red, named it "red", judged it more like orange than green, agreed that it was a "warm" rather than a "cool" color, etc., it necessarily had an experience of intense redness. Rather, the machine appears to simulate color experiences without actually having them. This difference between having and simulating experience underlies Searle's (1980) distinction between "weak AI" and "strong AI".

Figure 6

Even so, it is surprisingly difficult to prove that this machine fails to have color experiences. A card-carrying functionalist would claim that such a machine does have color experiences purely in virtue of the computations it performs. That may seem unlikely to readers not in the grip of functionalism, but can it be refuted? The underlying difficulty is the "problem of other minds." Because we do not have access to the experiences of any other entity -- be it a person, animal, or machine -- how can we tell whether the color machine actually has color experiences as a result of performing these computations? There seems to be a logical possibility that it might, but Searle (1980) has argued that it cannot. Let us therefore consider an adaptation of his argument to the present topic of color experience.

2.5 THE COLOR ROOM

Searle's original thought experiment, known as the Chinese room argument, did not concern color perception, but understanding foreign languages. He asked whether a computer could literally understand Chinese if it were programmed so that when it was asked questions in Chinese, it answered in a way that was indistinguishable from native speakers of Chinese. The claim of "strong AI" is that such a computer actually understands Chinese, including whatever conscious experiences go along with such understanding, but Searle has claimed it does not.

Searle's argument can be adapted to make a corresponding, and in some ways simpler and more compelling, point about the insufficiency of performing information processing operations to produce color experiences. Consider a system that takes as input the quantum catches of the three different types of receptors responsible for color vision (the cones sensitive to short, medium, and long wavelengths), and gives as output the name or description of the color in some language. The claim of strong AI is that such a system must be having color experiences rather than just simulating them. This is a form of functionalism because the claim is that if the machine's internal color states and processes are causally isomorphic to people's color states and processes, then the machine literally has the analogous mental states, including any experiential component.

The thought experiment is to imagine yourself inside a "color room." You receive an ordered set of three numbers through the "input" door. You then consult a rule book that tells you how to perform the various numerical calculations necessary to arrive at three other numbers (corresponding to hue, saturation, and brightness of a light, although you do not know that this is what they represent). Ultimately, the rules in the book tell you how to use these values to select one of a particular set of letter strings to hand out the output door. If the language of color naming is German, and if you do not know German, these would be meaningless strings of letters to you: "schwartz", "weiss", "rot", "grün", "blau", etc. You sit in the color room, carrying out your computations flawlessly, but totally mechanically, because nothing you are doing has any meaning -- other than the fact that you are performing various arithmetical operations and following certain logical rules that govern which letter strings you pass out the "output" door.

Unbeknownst to you, however, people outside the room interpret the three numbers you receive as inputs to be the quantum catches of the three types of color-sensitive receptors in the retina in response to a colored light. They also interpret the meaningless (to you) letter strings you send out as basic color terms in German. Thanks to the rules in the book, the letter strings you send out in response to the three numbers turn out to be the appropriate German names for the color patches that produced the three input numbers.

Let us suppose that you become so well practiced at your task that your performance is indistinguishable from that of German speakers with normal color vision. Let us also suppose, for the sake of argument, that the rules you are following in the book conform precisely to the established operations the visual nervous system uses to arrive at internal representations of colors and color names. These two facts mean that you have satisfied the usual functionalist criteria for claiming that you have mental states associated with color perception and naming. The key question is whether carrying out these computations would necessarily cause you to have the color experiences you normally would have when viewing the corresponding patch of color.

Most people's intuitions on this question are perfectly clear. Performing these computations, no matter how closely they might correspond to visual information processing in response to patches of color, would not give rise to anything like the experiences of color that would spontaneously arise if you were actually looking at the patch of light.

The force of this argument is that it avoids the problem of other minds in deciding whether a color machine has color experiences. By inserting yourself into the color room system of rules and computations, you know that there is a conscious agent within it who could have color experiences if that were indeed a necessary consequence of doing the right computations. Defenders of strong AI have attempted to refute Searle's argument on a number of grounds. We will not consider these arguments here as they are fairly involved and readily available in the commentaries following Searle's original article, together with Searle's responses to them. Their translation into the color room version is straightforward.

It appears that behavioral methods are not only deficient in determining the qualities of internal experiences, but in determining whether there are any internal experiences at all. They can determine the relational structure among experiences, but not their subjective quality, including whether they are simulated (i.e., nonexistent) or real. This appears to be a great deal less than one expects from a full scientific understanding of consciousness.

3 BIOLOGY TO THE RESCUE?

Having considered the inherent limitations of behavioral approaches to understanding color awareness, let us now consider the prospects for going further via biological approaches, as suggested by neuroscientists (e.g., Crick, 1994). Will physiology succeed in penetrating the subjectivity barrier? In particular, will it allow us to go beyond the isomorphism constraint to get a handle on the quality of color experiences within and across individuals? If so, how? And if not, why not?

3.1 CORRELATIONAL VERSUS CAUSAL THEORIES

In considering the status of physiological hypotheses about experience, it is important to distinguish two different sorts, that I will call correlational and causal claims. Correlational claims concern the type of brain activity that takes place when experiences are occurring and fails to take place when they are not. Claims that conscious experiences arise only when there is brain activity at a particular location, within synapses employing a particular neurotransmitter, or firing at a particular rate would all be correlational, because they provide no causal explanation of how this particular kind of brain activity produces experience. They fail to fill the "explanatory gap" between ordinary physical events and experience (Levine, 1983) because they merely designate a subset of neural activity with which consciousness is associated. No explanation is given for this association; it simply is the sort of activity that accompanies consciousness.

At this point it would be appropriate to contrast such correlational claims with a good example of a causal one: a theory that provides a scientifically plausible explanation of how a particular form of brain activity actually produces experience.{12} Unfortunately, no examples of such theories are at hand. In fact, to this writer's knowledge, no one has ever suggested any theory that the scientific community regards as giving even a remotely plausible causal account of how experience arises from neural events. This doesn't mean that such a theory is impossible in principle, but only that we haven't yet hit upon a serious candidate.

A related difference between correlational and causal biological theories of experience is that they are likely to differ in generalizability. Correlations are inherently specific to the particular biological system within which they have been identified. In the best-case scenario, a good correlational claim about the neural substrate of human consciousness might generalize to chimps and certain monkeys, possibly even to dogs or rats, but probably not to frogs or snails simply because their brains are too different. If a correlational analysis showed that human consciousness were associated with, say, GABA activity, would that necessarily entail that creatures without GABA were not conscious? Or might some other evolutionarily related neural transmitter serve the same function in brains without GABA? If so, what features might they have in common? Even more drastically, what about extraterrestrial beings whose whole physical make-up might be radically different from our own? In such cases, a correlational analysis is almost bound to break down. A causal theory of consciousness might have a fighting chance, however, because the structure of the theory itself could provide the lines along which reasonable generalizations might flow.

3.2 THE DNA ANALOGY

It seems conceivable that we will someday attain a full enough understanding of the brain that the biological mechanisms underlying conscious experience will be discovered and understood, even though we currently have few clues as to what they might be like. The best we can do right now is appeal to an analogy with the understanding of the nature of life that was achieved by discovering the molecular structure of DNA.{13}

The facts to be explained by a theory of life concern certain differences in macroscopic properties of living organisms versus nonliving objects, particularly their abilities to grow, sustain their internal functions, and reproduce. The basic mechanisms available for a truly causal theory to explain such life functions are the physical behavior of ordinary matter. Prior to the discovery of the structure of DNA by Crick and Watson, however, there was a huge explanatory gap between the known physical laws and these molar properties of living organisms. The gap was large enough that some theorists claimed there must be a special "vital force" that pervades living tissue and endows it with the requisite properties. Other theorists rejected this view, of course, insisting on a mechanistic theory devoid of mysterious, nonphysical forces.

In retrospect, the facts about living organisms that turned out to be most important for discovering the mechanisms of life concern their ability to reproduce. Especially pertinent were the lawful inheritance relations between characteristics of offspring and characteristics of their parents. The general nature of these regularities was worked out by the brilliant Austrian monk, Gregor Mendel, in his painstaking experiments with garden peas. His results enabled him to propose, test, and refine a theory of inheritance based on hypothetical physical entities, now called genes, that were combined in reproduction and expressed in offspring according to lawful principles. He worked out this genetic theory without ever directly observing the physical basis of these genes. Mendel's work is a beautiful example of how a molar behavioral analysis can give insights into underlying molecular mechanisms without those mechanisms being directly observed.

Mendel's genetic theory did not fill the explanatory gap, however, for the physical mechanism was not yet specified. Many years later, when sufficiently powerful microscopes enabled biologists to observe the internal machinery of cells directly, correlational hypotheses correctly suggested that the nucleus of cells was crucial in their self-replicating abilities. More refined correlational conjectures focussed on the strands of chromosomes within the nucleus as the crucial structures involved in cell reproduction. Even so, the explanatory gap remained, since there were no serious candidate theories about how the properties of living cells might arise from chromosomes, whose internal structure was unknown.

The gap-filling discovery was made by Crick and Watson when they determined the double helical structure of DNA. It revealed how a complex molecule, which was made of more basic building blocks, could unravel and replicate itself in a purely mechanical way. As further implications of its structure began to be worked out, it became increasingly clear how purely physical processes could form the basis of not only genetics, but cellular replication, protein synthesis, and a multitude of other unique and previously inexplicable properties of living tissue. The current biological understanding of life is thus a good example of how a truly causal theory can fill a seemingly immense explanatory gap in science. It did not appear magically, but was historically preceded by molar analyses of organismic phenomena and by correlational observations of the underlying biological mechanisms. Indeed, if Crick and Watson had worked out the structure of DNA prior to these other discoveries, its importance would very likely have gone completely unnoticed. Thus, all levels of theorizing were important in the historical development of a scientific theory of life.

In this analogy, our present knowledge of the biological substrates of conscious experience is best thought of as pre-Mendelian. After decades of actively ignoring the problem, scientists of many persuasions are beginning to work on it. Behavioral scientists are looking for functional correlates of consciousness in the hope of discerning regularities that would enable the formulation of computational theories (e.g., Baars, 1988; Johnson-Laird, 1983; Marcel, 1983). Such theories can be thought of as akin to Mendelian genetics: hypotheses about the nature of consciousness at an abstract information-processing level distinct from the actual physical mechanisms that produce it.

Biological scientists are also beginning to look seriously for neural correlates of consciousness in the hope of narrowing the problem to some relevant subset of the brain where conscious experiences arise (e.g., Crick, 1994; Crick & Koch, 1995; Sheinberg & Logothetis, 1997). This work can be likened to microscopic studies that identified the special importance of the nucleus and particularly of the strands of chromosomes in cellular division and reproduction. This biological analysis of the correlates of consciousness is barely in its infancy, and much work will be required before we know where to look for the relevant mechanisms. And we are still very much in the pre-DNA phase of our quest to provide a causal explanation of conscious experience. Nobody yet has a clue about what such a theory might look like. When and if it is discovered, it will be a scientific breakthrough of staggering importance and implications, for it will unlock one of the deepest scientific problems of all time. Many scientists believe this to be possible, perhaps even likely (e.g., Crick, 1994). It may not turn out to be a purely reductionist account, as the explanation of genetics by DNA turned out to be, but most believe that there is a biological explanation and that it will eventually be found.

But even if this is true, it is still unclear whether biological science will be able to take us beyond the limitations of behavioral science in understanding the nature of experiences themselves. By analogy with the DNA story, the first task surely is to find out what aspects of brain activity correlate with different experiences. Then we can ask whether this will tell us more about underlying experiences than we can find out behaviorally. Finally we can ask about the prospects for achieving a truly causal explanation.

3.3 NEURAL CORRELATES OF COLOR EXPERIENCE

Many questions about the neural correlates of color experience can be asked and answered within a biological framework. Some already have been, using present knowledge and technology. We know, for example, that the neural mechanisms underlying color experience must be somewhere in the cortex rather than in the retina or precortical visual system. One line of evidence is the existence of a form of color blindness or color weakness, called achromatopsia, that is caused by damage to a certain region of the visual cortex between the occipital and parietal lobes of the brain (Meadows, 1974). We do not yet know whether this region is actually the neural locus of color experiences or whether they occur further along in the chain of neural processing, but we are slowly working our way toward this goal. It may involve activity in the frontal lobes, as Crick and Koch (1995) have suggested for consciousness in general, activity in the anterior cingulate gyrus, as Posner and Raichle (1994) have proposed, or activity in some other, as-yet-unsuspected structure in the brain. In any case, there seems little reason to doubt that someday the region(s) of the human brain in which color experiences arise will be discovered and the pertinent properties of neural activity there will be identified. The question I now want to address is whether such discoveries are likely to tell us any more about color experience than we already know from our own experiences and from the results of behavioral studies. In asking this question, there is no reason for us to restrict our inquiries by the technologies presently available, so from here on we will resort to thought experiments in which we are freed from such limitations.

Let us first consider how correlations between brain events and color experiences could be studied and what might be discovered by doing so. Suppose I am a subject in an experiment in which the activity of every cell in my brain can be monitored by some futuristic "brainometer." Neuroscientists could then study the differences between various forms of conscious and unconscious activity in my brain by correlating this neural activity with my color experiences. They could find out, for example, what particular patterns of neural activity in what particular regions of my brain correspond to my experiences of particular shades of red, orange, green, or any other color.

Presumably, the structure of these patterns of neural firings would turn out to be isomorphic to the structure of my color experiences and also to their representations in some suitably structured color space. This would mean, to take just one example, that the pattern of neural activity corresponding to red experience -- whatever it might be -- should be more like the pattern corresponding to orange experience than to the pattern corresponding to green experience. This is just the relation envisioned by Gestalt psychologists in their doctrine of psychophysiological isomorphism (Köhler, 1929). I see no conceptual problems in carrying out this program of research -- other than inventing the brainometer, of course. The results would tell us a great deal about the neural correlates of color experiences. But would they allow neuroscientists to determine the quality of my color experiences or even to determine whether my color experiences are the same as yours?

Notice that although these correlations are intended to study relations between my brain's neural activity and my qualitative experiences, they cannot be computed from my experiences themselves, because I alone have access to these experiences. Rather, they would have to be computed from records of my behavioral reports of my experiences, because calculating conditional dependencies between two phenomena (in this case, patterns of brain activity and color experiences) requires repeated, objective, quantitative measurements of both. Since my experience can only be objectively (and incompletely) assessed through my behavior, the correlations will actually reflect dependencies between my brain activity and my behavior. That being the case, it is unclear how such a correlational approach will be able to go beyond the limitations of behavioral science in establishing the nature of color experiences, since behavioral methods are inherent within it.

This is not to say that we will fail to learn anything about the biology of color experience from the brainometer experiments. Far from it. We will be able to learn everything there is to know about the biological correlates of experience up to the level of isomorphism. The results would tell us, for example, what the neural correlates are for hue, saturation, and lightness, or for redness/greenness, blueness/yellowness, and blackness/whiteness.{14} But the correlated brain activity cannot inform us about the quality of the experiences themselves, as they still might be any set of experiences that have the required relational structure. We have hit the isomorphism constraint again.

3.4 DEFINING NEUROLOGICAL EQUIVALENCE CLASSES OF COLOR EXPERIENCE

What about the more modest goal of using biological methods to settle the color transformation problem? Even if scientists cannot determine the exact quality of my (or your) color experiences from their neural correlates, can they at least determine whether our experiences of colors are the same? This is a weaker problem for it only requires determining whether you and I belong to the same equivalence class of color experiencers rather than determining what our experiences are actually like.

The obvious approach to this issue is simply to have neuroscientists, armed with their brainometers, correlate your brain activity with your reports of color experiences under controlled conditions, just as they did for mine, and then compare the two patterns of activity. Certain relations between our patterns of brain activity might superficially suggest certain specific kinds of color transformations. For example, less extreme firing rates in opponent process color cells in your brain might suggest a contraction of color space, or the reversal of firing rates in such cells of your brain might suggest an inversion transformation. But only if your brain activity were identical to my brain activity in response to the same stimuli would one be justified in concluding that our experiences of the colors were necessarily the same. If they were identical, any other conclusion, however logically valid, would be eliminated by Ockham's razor as unparsimonious, for in the absence of any difference in brain activity, it is simpler to suppose that our experiences are the same than that they differ in some mystical, ineffable way. This conclusion is consistent with the philosophical notion of supervenience (Kim, 1984).

Assessing whether our patterns of brain activity are the same is not as straightforward as it might seem, however, for it requires specifying a principled physical correspondence between our two brains. If those portions of our brains that are involved in color perception were identical, establishing this correspondence would not seem to pose much of a problem.{15} This situation might arise for a tiny subset of individuals, such as clones and identical twins, but the fact of the matter is that most people's brains differ from each other in a multitude of ways. Even seemingly minor physiological differences in color-relevant neurons -- such as variations in baseline firing rates, the fine temporal structure of neural spiking, the conduction speed of axons, or whatever -- might conceivably cause nontrivial differences in experience. In the absence of a causal theory of experience, it would be impossible to know which differences matter and which do not.

Even this problem of neural variation might not prove insurmountable if there were some way of studying the relation between biological differences and experiential differences. That would enable scientists to determine whether variations across individuals in measures of baseline rates, fine temporal firing pattern, axon conduction speed (or whatever) systematically influences experience. But here we run into the subjectivity barrier again, because assessing experiential differences between individuals requires somehow comparing their color experiences. As we have seen, such comparisons below the level of isomorphism are defeated by the inherent subjectivity of experience and its underdetermination by purely behavioral methods.

If there are objective differences in the relational structure of our experiences, such as occur in the various kinds of color blindness, appropriate behavioral methods can clearly detect them, and those differences can then be correlated with differences in brain activity. This enables us to determine a set of behaviorally defined equivalence classes of color perceivers -- normal trichromats, protanopes, deuteranopes, etc. (see Figure 7) -- according to the relational color discriminations they can make. But if our relational structures are identical, then our experiences are still free to vary within the isomorphism constraint without detection. And if there are any potentially relevant differences between our brains that might produce experiential differences, it is unjustified to assume equivalence of color experiences. Exactly how one decides what kinds of differences between neural events are "potentially relevant" to experience is a serious problem, but one to which there are potential solutions, at least in principle (see below). Many factors will presumably be judged irrelevant by these criteria, but others are likely to remain.

Figure 7

It is important to note that even if we were able to conclude that two individuals' experiences are the same because their underlying physiology is the same, this would not specify the quality of their experiences to an outside observer. It would indicate only that, whatever they might be, they are the same. Only if I am one of the people whose experiences have been determined to be the same (or, from your perspective, if you are one of them), will this fact determine for me (or for you) what the experiences of another person are actually like. In other words, valid criteria of experiential sameness based on sameness of physiology would allow one to establish equivalence classes of experiential qualities, but it would determine the quality of those experiences only for the particular class of which the self is a member. An important question remains, however: How are we to determine which physiological differences matter for color experience?

3.5 WITHIN- VERSUS BETWEEN-SUBJECT DESIGNS

We encounter the subjectivity barrier every time we try to make comparisons of subisomorphic differences across individuals. It therefore seems clear that progress in identifying experiential differences below the level of isomorphism can only be made by studying the problem within individuals. If pharmacologists were able to produce color-altering drugs, for example, their experiential and physiological effects could be studied and correlated within individuals. If administering "invertacillin," for example, produced a red-green reversal of color experiences within individuals, then some combination of its various physiological effects must have produced this change. Scientists would have to rule out certain bizarre, but logically possible, changes in language or memory as alternative explanations, but we will suppose that such alternative explanations can be effectively eliminated.{16}

It seems intuitively plausible that subisomorphic changes in experience due to physiological interventions would be detected in appropriately designed within-subject experiments. We naturally suppose that if the experienced world changed color over a period of seconds, minutes, or even hours, we would notice that this had happened and in what way it had changed. Surely one would realize that grass now looks red instead of green, and stop signs green instead of red. Even so, various philosophers have tried to imagine scenarios in which such changes would not be detectable within an individual (e.g., Putnam, 1965), but the conditions required are complex, convoluted, and generally unconvincing (e.g., Dennett, 1991). The difficulty, in a nutshell, is that although changing the color experiences would be (conceptually, though not practically) rather simple, they could easily be detected through mismatches with memories, such as noticing the oddity of seeing red grass. This is precisely the kind of change in experience that would allow subjects in our hypothetical experiment to report that the world looked different after the physiological intervention.

In fact, there is clear evidence that people can detect changes in color experience under circumstances similar to those we are suggesting. Strokes that damage particular areas of visual cortex (in the occipito-parietal area) result in spontaneous reports by patients that their color experience has disappeared or has lessened noticeable in intensity (Meadows, 1974). Reports from these patients with achromatopsia thus provide evidence that changes in color experience can be detected within individuals by this means of memory comparisons in at least some cases. There is also evidence that other types of color changes go experientially undetected, however. Hardin (1990) has argued that because the lens yellows with age, the precise colors a person experiences will change, albeit glacially, during their lifetime, a change that nobody seems to notice. This latter example is less relevant to our present concerns, however, because the changes in color experience we are contemplating are both swift and enormous compared with the slow, slight yellowing of the lens. Still, it is worthwhile to realize that we are making some assumptions in supposing that people will be able to report changes in color experience over time as a result of a biological intervention.

One important use of within-subjects experiments is to determine whether or not certain physiological differences are relevant to color experience. If color experiences do not change when an intervention alters some aspect of an individual's neural structure or function, then that particular aspect can be ruled out as responsible for differences in color experience, at least in that individual. The situation is actually a bit more complex than this, because even if a given factor does not influence color experience, it might have some influence in conjunction with another factor. The set of potentially relevant variables is thus the orthogonal combination of all possible single factors, which is presumably a very large set. In any case, each factor or combination of factors that can be eliminated enlarges the number of individuals in each neural equivalence class, and thus increases the number of individuals who can be inferred to share a particular set of color experiences at the subisomorphic level.{17}

One of the most interesting findings about the subjective quality of color experiences comes from a within-subject design, albeit one due to nature rather than scientific intervention. The question it addresses is how colors appear to colorblind individuals versus people with normal color vision. The subjectivity barrier thwarts direct, between-subject comparisons of experiences, even though we can obtain good behavioral evidence that people differ in the particular pairs of colors they can discriminate. But a within-subjects comparison is possible for individuals who have one color-normal eye and one color-blind eye (Sloan & Wollach, 1948; MacLeod & Lennie, 1976). Although such people are extraordinarily rare, they report that some colors in their color-blind eye look essentially the same as in their normal eye, that others appear uncolored (gray) instead of colored, and that still others appear to be mixtures of grays and colors. Through a red-green colorblind eye, for example, such subjects report seeing everything in various shades of blue, yellow, and gray, as if the normal color solid were projected onto the blue-yellow-black-white plane of color space. This result is not definitive for the more general, between-subjects case because the two eyes of such a subject are connected to the same brain, and this may strongly constrain color appearances arising from stimulation in the colorblind eye. Even so, it is an interesting and important finding, one that is only possible because nature has provided an unusual within-subject design.

Many of the fascinating phenomena responsible for the recent surge of interest in consciousness can also be cast within this within-subjects framework. One of the classic examples is blindsight, in which a biological intervention (surgical removal of occipital cortex in one hemisphere) caused an individual to fail to have experiences of the sort he used to have in half his visual field (Weiskrantz, 1986). Knowing full well what it was like to have visual experiences in his left visual field before the operation and what it was still like to have them in his right visual field, D.B. enabled scientists to conclude that something about the surgery had eliminated visual experience and gave impetus to the search to find out what was responsible. Unilateral neglect and other syndromes of organic brain damage tap into the same format of within-subject designs with biological interventions induced by natural causes. They are among the most powerful tools in current scientific methodology for exploring biological substrates of experience.

Thus far, we have only considered correlational studies of experience and argued that, unless the color systems within two brains are identical so that their experiential content can be assumed to be the same, comparisons between color experiences across individuals will be thwarted by limitations inherent in behavioral science. But what about a causal theory of consciousness? Might not such a theory allow between-subject comparisons of color experiences to be made directly? Might not such a theory even be able to determine the quality of different people's experiences of color in an objective manner?

Because we have no idea what a causal theory of consciousness might be like, it is impossible to determine whether it will contain the sort of mechanisms from which one could predict the specific quality of color experiences within and/or across individuals. For the sake of argument, though, let us suppose it purports to do so. A nontrivial further requirement of scientific theories, however, is that they be testable. Could a causal theory of the qualities of experience be tested?

Again the answer depends on whether the kinds of variations it predicts can be induced within individuals or only across individuals. If a physiological intervention -- be it a drug, surgical procedure, or brain-controlling machine -- can be identified that will reliably alter the crucial neural properties in the required ways, then it could be administered to individuals and its effects assessed by having subjects compare their color experiences before versus after the intervention. If colors became systematically lighter, or if redness/greenness reversed, or if colors began to be experienced as sounds, as predicted by some causal theory, then it would be supported; otherwise not. Although such experiments might be technically difficult, they are conceptually straightforward.

If the required changes could not be induced within individuals, but only examined across them, we run into the subjectivity barrier again. Differences that can be measured behaviorally -- which I claim reflect differences in relational structure -- can of course be detected and compared with the predictions of the theory. But if the theory makes predictions that certain physiological differences will cause experiential differences across individuals below the level of isomorphism, they cannot be objectively tested. Biological methods can verify that the appropriate difference is present across individuals, of course, but behavioral methods will not be able to detect its subisomorphic effects on experience for the reasons we have already discussed.

The advantages of within-subject manipulations in revealing subisomorphic variations in color experience actually fit rather well with the intuitions on which the inverted spectrum argument is based. The reason inverting the spectrum seems so compelling is that I can easily imagine what it would be like for me to experience such a transformation of color experience. If I awoke one morning to find my color experiences transformed, I believe I would know immediately that it had happened and how it was different. This is essentially the within-subject design I have argued is the only method with which one might be able to break the subjectivity barrier. And it wouldn't actually "break" it as much as merely avoid it because, within an individual, there is no subjectivity barrier.{18}

But even within-subject experiments have serious limitations arising from the subjectivity barrier. They may enable scientists to detect experiential differences within individuals over time that cannot be detected across individuals, but they still cannot fix the quality of particular experiences. The examples we have discussed only allow scientists to conclude that the experiences changed in the particular way described by the subject, not how they were before the change nor how they are after it. That is, an external observer can only find out about the relation between the experiences before versus after the intervention. This does not allow any inferences about the actual qualities of either set of experiences. For example, if some drug produced a red/green reversal in a subject, that fact could be determined, but it would not constrain the nature of either green or red experiences in any way. This difficulty should sound familiar, because it is the isomorphism constraint again. It arises because the subjectivity barrier is still in place: not between one subject and another, but between the subject and the experimenter And it will always be there as long as scientific methods require objective measurement.

The theory of mind most relevant to the considerations raised in this paper is functionalism because of the central role relations among mental states play in both cases. In particular, functionalism identifies the nature of mind with the relations among mental states and their causal connections to the environment. Crucial to the functionalist view is the assumption that the nature of mental states does not depend on their particular physical realizations. Minds must be physically embodied somehow in order to have the necessary causal connections to the world, of course, but functionalists claim that the same mind could be instantiated in wildly different hardware, neural or otherwise, as long as the same functional relations are present. These claims have sparked heated arguments about the adequacy of functionalism as a metatheory of mind, most of which seem to be aimed at convincing the reader either that functionalism is obviously right or obviously wrong. The considerations raised in the present paper suggest that a better articulated critique is possible.

Using color experience as a paradigm example, I have argued that there is a metaphorical brick wall that limits the kind of knowledge about experience that can be derived from behavior. On one side of this epistemological wall lie objective facts about relations among experiences that are part of our shared cognitive culture and that can be studied behaviorally. On the other side lie subjective, private aspects of individual experiences that cannot be. Theoretically, I have identified this limitation with the isomorphism constraint, claiming that behavioral science can specify the structure of experience up to the level of isomorphism, but no farther. This analysis is intimately related to functionalism because it states that what can be known behaviorally are relations among experiences. Such relations are, by definition, within the domain of functionalism, and so anything that can be known about experience from behavior (i.e., up to the isomorphism constraint) can be captured by a functionalist account. I have also argued, however, that the nature of individual experiences (beyond the isomorphism constraint) cannot be known from behavior, even in principle. These nonrelational aspects of experience lie, by definition, outside the domain of functionalism for they are underconstrained by relations among mental states. It seems that the failure of functionalism to provide an account of these aspects should be counted against its claim of fully specifying the nature of mind. It doesn't seem to be able to do the whole job.

But it can do at least part of the job. Specifying conscious mental states to the level of isomorphism is nothing to be sneezed at. In the domain of color experience, it is sufficient to completely specify the shape and structure of color space, which includes everything we know about color experience, scientifically speaking. There is even a serious question about whether there is anything more that can be known within the methodological/epistemological limitations of science. I have argued, for example, that no theory of consciousness, even a causal biological one, will be able to produce a testable account of the nature of individual experiences because the behavioral aspects of the required tests would not support inferences beyond the isomorphism constraint. One might argue, therefore, that it is unfair to require of functionalism (or any other theory of mind, for that matter) that it be able to account for individual experiences, since such theories cannot be tested in any case.

The one way in which I have argued that we may be able to get some understanding of experiential qualities beyond the level of isomorphism is by the sort of "end-run" I described previously. If within-subject experiments using biological interventions can enlarge the experiential equivalence class of color perceivers beyond individuals (e.g., just myself) to groups of individuals (e.g., you and me), then I can infer that anyone within my equivalence class has the same color experiences I do. Such equivalence classes lie beyond the isomorphism constraint because there could, in principle, be many of them within a population of perceivers who are behaviorally indistinguishable (see Figure 7). Science could not objectively specify what the nature of the underlying experiences are within any of these equivalence classes, but I would have subjective knowledge for the experiences of members of my own equivalence class, just as you would for members of your own equivalence class. So it seems that functionalism may succeed in specifying experience as fully as objective science will allow, even though I, as a conscious experiencer of colors, may be able to get some further knowledge via attributing my own subjective experiences to my equivalence class as defined by combined evidence from the brain and behavioral sciences.

The indeterminacy of the nature of individual experiences within functionalism is a small shortcoming, however, compared with the fact that it fails to discriminate between my experiences and the complete lack of such experiences in an information processing system that has the same causal relational structure among its processes, but no experiences whatever. The color machine described above, for example, has the same representational color space as a normal human trichromat, but, because of the implications of the color room argument, it seems highly unlikely that such a machine would have color experiences of any sort. The causal isomorphism of its color representations to those of normal trichromats is sufficient to guarantee that it cannot be distinguished from a normal trichromat by behavioral means, but not that it has color experiences of any sort. This is a problem because it means that, in addition to having no account of the qualities of experience, functionalism has no account of experience at all.

There is an interesting parallel here between implementation and experience with respect to functionalism.{19} It has been recognized from the outset that functionalism treats mental phenomena as independent of their physical realizations: any set of physical events will do, provided they have the right causal relational structure. One way of stating this situation is that functionalism induces a set of equivalence classes on objects in which all those with precisely the same causal relational structure among their internal states fall into the same equivalence class, despite potentially enormous differences in their physical implementations. A similar claim can be made about functionalism in the experiential domain. Functionalism appears to be independent of the experiential realization of mental phenomena in much the same sense: Any set of experiences will do -- indeed, even nonexperiential computational states will do -- provided they have the right causal relational structure. Functionalism thus establishes a set of equivalence classes on minds in which all those with the same causal relational structure among their experiences fall into the same equivalence class, despite potentially enormous differences in their experiential realization.

Notice a crucial difference between these two cases, however. Although functionalism asserts that minds are independent of their particular physical realization, it maintains that they must have a physical realization to meet the requirement of proper causal connections to the physical world. Functionalism is therefore merely indifferent about what particular implementation it has. But in the corresponding case of experiential realization, functionalism does not demand that minds have one at all. There is no requirement corresponding to the causal connection to the environment that can be used to enforce some kind of phenomenal component as there is in the case of a physical component. Experience thus does not have any intrinsic necessity within a functionalist framework, even though experience is the defining characteristic of mental life.

The conclusion I draw is that functionalism is the appropriate, state-of-the-art theory of mind from the standpoint of purely behavioral science, but it falls short of providing a full account of mental events. This is perhaps not too surprising given that one of the cornerstones of functionalism is the irrelevance of physical implementation. Without physiology, what objective facts are there to constrain the nature of mind except behavioral ones? And if behavior can only reveal relational aspects of mental events, then a theory of mind built solely on relations can be expected to do as well as can be done. Within-subjects designs employing physiological interventions, natural or artificial, provide ways of going beyond these limitations, however. If such research is successful -- and neuropsychological phenomena such as blindsight suggest that it will be -- more comprehensive, physically based theories of mind will be required to account for the fact that certain changes in neural realization produce behaviorally detectable changes in experience within individuals.

The considerations discussed in this article cast serious doubt on the possibility of science being able to give a complete and testable explanation of the quality of color experience, or any other kind of experience, for that matter. Behavioral science is limited by the isomorphism constraint in making between-subject comparisons: it cannot detect differences in the experiences of two individuals below the level of equivalent relational structure. Biological science is subject to the same limitation because behavioral measurements are the only way it can compare experiences across individuals. The prospects for going further using biological techniques depend on (a) establishing the identity of underlying physiology to a level at which it becomes implausible to believe that experiential differences would arise from such minor physical differences and/or (b) employing physiological interventions using within-subject designs. Even if both conditions can be met -- and each will be difficult -- there is no way to uniquely specify the qualities of particular experiences except by reference to one's own. Physiological identity can only demonstrate that two people's experiences should be the same, not what actual qualities they have in common. Within-subject designs can examine changes in experience, but cannot reveal what they changed from or to. Thus, explaining the qualities of individual experiences may be one mystery that will not yield its secret to the seemingly inexorable advance of science.

Acknowledgement

Requests for reprints should be sent to Stephen E. Palmer at the Psychology Department, University of California, Berkeley, CA 94720-1650. The preparation of this article was supported by Grant 1-R01-MH46141 from the National Institute of Mental Health to the author. I wish to thank Alison Gopnik, Paul Kay, Eleanor Rosch, John Watson, John Searle, Bruce Mangan, Max Vellmans, Bernard Baars, Elisabeth Pashiere, C. Lawrence Hardin, Robert E. MacLaurey, and two anonymous reviewers for their helpful comments on earlier drafts of this article.

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FOOTNOTES

{1} Although there is a great deal of consensus among color scientists about the primacy of these four chromatic colors, the matter is not completely settled. Indeed, Saunders and van Brankel (1997) have disputed various aspects of this idea in a previous article in this journal, to which the interested readers is referred.

{2} Many people object that green is not a primary because it is a mixture of blue and yellow. Although it is true that mixing blue and yellow paints or dyes generally produces some shade of green, people without experience mixing paints do not report that green looks like a the composition of blue and yellow. In any case, virtually everyone given control over the degree of yellowness/blueness in a green test patch can easily find a setting where it looks neither yellowish or bluish. This color is unique green.

{3} The color space in Figure 2 is intended to be an abstract schematic model of human color experience rather than a particular one, such as Munsell space, OSA space, NCS space, CIE space, etc., each of which have somewhat different properties in terms of both the types of stimuli and nature of the relations they represent. This model will be used to explicate a mode of argumentation with respect to empirical constraints on the inverted spectrum argument rather than to demonstrate definitive results concerning fine points in terms of which alternative models of color differ.

{4} Again, this generally accepted view of the dimensionality of color experience is not beyond reproach. Saunders and van Brakel (1997) have disputed that these are the only or the most appropriate dimensions of color experience, citing a number of dissenting voices in the research literature. The interested reader is refered to their article for further discussion.

{5} It can be objected that there is no reason to suppose that such individuals are any different from normal trichromats in any biologically meaningful way. This would be true, however, only if there is no difference between L- and M-cones except the nature of the pigments they contain and the genetic codes that determine those pigments. If there are any other differences between L- and M-cones that are determined by other genes -- e.g., their frequency of occurance, retinal distribution, or output relations to other cell types -- then such individuals should not be biologically lumped with normal trichromats.

{6} This presentation reflects what I take to be a plausible first-order story about color categories, one that may or may not be correct in detail. I present it to illustrate the logic of the claim that the structure of color categories breaks further symmetries of an empirically accurate color space rather than as a definitive argument that they do so. Many aspects of the Berlin-Kay-McDaniel theory I discuss are open to question, as Saunders and van Brakel (1997) have argued at length.

{7} The status of goluboi as a BCTs is controversial, with some researchers endorsing its inclusion, and others not (see MacLaury, 1997).

{8} Tarski (1954) took a set theoretic approach to models by defining an n-ary relation among objects in terms of the set of ordered n-tuples of objects for which that relation holds. For instance, the binary lighter-than relation between two color experiences, lighter-than (ci, cj), can be defined as the set of all ordered pairs of color experiences <ci, cj> such that ci is lighter than cj. This relation, in the sense of a set of ordered pairs, is then structurally preserved by the binary higher-than relation in a spatial model of colors if there is a function, F, that maps colors (c1, c2, ... , cn) to points (p1, p2, ..., pn), F(ci) = pi and F(cj) = pj, such that <pi, pj> is a member of the higher-than set of ordered pairs of points if and only if <ci, cj> is a member of the lighter-than set of ordered pairs of colors. These conditions ensure that the structure of one set of objects and the relations among them is identical to the structure of another set of objects and the relations among them, without either the corresponding objects or relations being identical in any sense whatever. (In the discussion of interobserver color transformations, the relevant mapping is from color experiences in one observer to color experiences in another observer.)

{9} Formally, an isomorphism can be defined as a function, F, that maps a relational system, S = <E, R1, R2, ... Rn>, onto another relational system, S' = <E', R1', R2', ... Rn'>, such that <ei', ej'> is an element of Rk' iff <ei, ej> is an element of Rk and F(ei) = ei' and F(ej) = ej'. E and E' are simple sets of elements in the two relational systems (in this case, color experiences of two different observers), and Ri and Ri' are sets of n-tuples that constitute relations among the elements within the two relational systems (in this case, relations among color experiences of two different observers).

{10} Transitivity requires that if A is isomorphic to B, and B is isomorphic to C, then A is isomorphic to C. Symmetry requires that if A is isomorphic to B, then B is isomorphic to A.

{11} The concept of isomorphism has a long and somewhat confusing history in psychology. Members of the Gestalt school discussed two different relations in terms of isomorphism (e.g., Köhler, 1929; Koffka, 1935). The clearer and less problematic use was to describe the "psychophysiological" relation between sensory experiences and underlying neurological processes in terms of having the same abstract structure. (We will have more to say about this relation later in this article.) The other was to describe the relation between the physical world and underlying neurological representations. Here they introduced some confusion by implying that there was actual similarity between certain spatial qualities of a stimulus, such as a square, and the corresponding pattern of neurological activity, which might itself be similar to a square. A more realistic view of this psychophysical form of isomorphism was developed by Shepard and Chipman (1970) as a natural extension of Shepard's pioneering work in nonmetric multidimensional scaling (Shepard, 1962a, 1962b). Shepard and Chipman called this relation "second-order isomorphism" to emphasize that, unlike some of the Gestalt ideas about isomorphism, the "first-order" properties of representations do not have to be the same or even similar. The internal representation of a square thus does not have to be itself square or even remotely similar to a square, but, whatever it is, it must be more like the internal representation of a rectangle than that of a triangle or circle. (Strictly speaking, the "second-order" qualifier is unnecessary because mathematical isomorphisms are, by definition, abstracted from the "first-order" properties of their individual objects and concern only "second order" relational structure). A particularly clear and cogent early discussion of isomorphism in sensory psychology can be found in Hayek (1952, pp 37-41). None of these uses of isomorphism is the same as the present one, however, which concerns the relation between the experiences of two different observers who may, in fact, have qualitatively different experiences, but the same relational structure over those experiences.

{12} By "causal" I do not mean a theory of consciousness that specifies the nonconscious neural events in the causal chain that ultimately leads to consciousness. In color perception, for example, it is clear that the registration of wavelength information by activity in the short-, medium-, and long-wavelength cones of the retina is not itself conscious, but does lead to color experience somewhere later in the sequence of neural processing. I take it that the specification of this causal chain of neural events leading to consciousness is conceptually unproblematic, although the details are not yet understood.

{13} I hasten to add that this analogy is quite imperfect. I particularly do not want to be construed as claiming that consciousness must be reduced to some causal physical mechanism -- although it might -- in the same way that biological processes can be reduced to physical processes involving DNA or that consciousness is like the "vital force" which will be eliminated from scientific vocabulary once it is properly understood in mechanistic terms. Both of these conjectures are up for grabs. I offer it only as a particularly clear example of the important difference between correlational and causal theories in a scientific domain.

{14} There appears to be a fairly widespread belief that the neural correlates of experienced redness/greenness, blueness/yellowness, and blackness/whiteness are already known from single cell recordings of opponent cells in the retina and lateral geniculate nucleus of monkeys (e.g., De Valois & Jacobs, 1968). Although these results are indeed suggestive, they do not support such an extreme conclusion because it is extremely unlikely that the neural correlate of color experience arises in precortical structures. The most that can be said is that there may be some later representation of experienced color that has a similar neural coding in terms of three opponent processes, as Hering (1878/1964) suggested from a phenomenological analysis.

{15} We would not require that our entire brains be identical, for there may be many brain differences that would be irrelevant to our experiences of color: e.g., those reflecting different memories of autobiographical events, different motor skills, different likes and dislikes, etc. Our own experiences of color do not appear to vary over the course of our lives as these and many other factors change, although such conclusions rests on assumptions that can be challenged.

{16} For example, the red-green dimension of color experience might not have been reversed by the drug, but rather the set of associations to the relevant linguistic terms for reporting them. Grass would then still look green, but the observer would call it "red". Such differences could be ruled out in a number of ways. For one, observers could be shown normal and color-inverted pictures of objects with characteristic colors (green grass and red grass, red stop signs and green stop signs) and then be asked (a) which were normal and which color inverted and (b) the name of the color of the object shown. If the experiences were inverted, subjects would always be wrong in both tasks, whereas if the linguistic labels were inverted, they would name the colors incorrectly but discriminate normal from reversed colors correctly. Another approach would be to determine which physiological structures were altered by the drug. If they were in the color pathways of the visual system, color experience would be the more plausible alternative, whereas if they were in the linguistic or memory centers, a change in language would be more plausible.

{17} There is the further methodological difficulty in this program of research of accepting the null hypothesis. That is, to eliminate a given neurological factor as relevant to color experience, one has to claim that it has no effect, and this is statistically problematic.

{18} This is true for normal people under normal circumstances. However, in a variety of dissociative states -- such as during hypnosis, fugue states, and episodes of multiple personalities -- there appear to be subjectivity barriers within individuals. These are complex and interesting phenomena that will ultimately prove to be of great importance to understanding consciousness, but are well beyond the scope of the present discussion.

{19} I thank Alison Gopnik for pointing out this parallelism when we discussed an earlier draft of this paper.