Jean-Louis Dessalles
email: dessalles@enst.fr-

Qualia and Spandrels: an Engineering Perspective 1

Keywords: qualia, phenomenal consciousness, evolution, epiphenomenon, qualitative spaces

If we consider that cognitive abilities evolved to perform sophisticated processing of sensorial inputs and allow for better adaptive behavioural responses, then it seems that there is no functional need for the qualitative aspects of subjective experience. Qualia have been shown to be dispensable when causal explanations of behaviour are requested. They look very much like epiphenomena. Though qualia are private experiences, the structure of qualitative spaces for various sensory modalities can be investigated. Surprisingly, what such studies reveal is inconsistent with epiphenomenality. Qualitative spaces offer a familiar look to the eye of engineers. There is some evidence that qualia are optimally organised, in each modality, for functional discrimination. It is thus reasonable to hypothesise that such a design was produced by natural selection. If so, we must admit that qualia play a causal role in cognitive processes, even if such a causality still lies beyond our understanding.

Introduction

1. Qualitative properties are the most immediate aspect of our conscious experience. Familiar examples are pain, pleasure, redness, the taste of red wine. Phenomenal consciousness 2 is a basic component of our perceptual and proprioceptual experience. As such, it can be considered as a biological property of our species. We may attribute to some non-human species a similar ability to experience smell, colour or pain, even if their experience may be qualitatively different from ours. An obvious question then arises: why did phenomenal consciousness emerge in the phylogenetic history of these species?

2. Such a question may seem alternatively pointless or hopeless, depending on how it is understood. It is pointless if phenomenal consciousness, as a scientific object, is on principle _______ 1 This research would not have existed if Tiziana Zalla had not introduced me to the problematics of consciousness. I thank her and Georges Rey for their fruitful remarks and critiques on a first draft of this paper. I also thank Rafael Núñez for his friendly and helpful comments. I thank Joëlle Proust, Jose Luis Bermudez and Thomas Polger for their numerous suggestions that helped me improve the quality of this paper.

2 Following [START_REF] Block | On a confusion about a function of consciousness[END_REF], we need to distinguish here phenomenal consciousness from other cognitive processes, from self-consciousness and from higher-order conscious states. The feeling of being a single entity, the fact that some recalled events look familiar, the feeling of "ownership" about our mental states, the firstperson point of view, the ability to observe aspects of our cognitive functioning are other important features of what is called "consciousness". Nevertheless, all of them might be different aspects of consciousness, each one might be related to different functions and may eventually call for different accounts [START_REF] Zalla | Unité et multiplicité de la conscience : une étude critique des théories contemporaines à la lumière d'une hypothèse modulariste[END_REF].

Technical Report ENST 2001-D-012 Paris, 2001 Available at: http://www.dessalles.fr/papers/Dessalles_01082301.pdf reduced to structural or functional properties of the brain. We assume the traditional view that there is something like experiencing pain or redness, and that such qualitative experience is worth scientific explanation. We use the term qualia to refer to qualitative aspects of experience, being aware that the cognitive nature of qualia is highly problematic. The main claim of this paper should be easily transposed in any non-eliminativist theoretical framework about consciousness 3 .

3. Accounting for the very existence of qualia may seem rightfully hopeless, at least in a foreseeable future. It has often been stressed that we have no idea why qualia are what they are, why they are simply felt [START_REF] Chalmers | Facing up the problem of consciousness[END_REF][START_REF] Harnad | No Easy Way Out[END_REF]. The problem addressed in the present paper is much less ambitious. We take phenomenal experience as granted, and acknowledge the fact that its very existence remains mysterious. Given that, we question the possibility that phenomenal experience be considered epiphenomenal.

Are qualia mere evolutionary epiphenomena ?

4. Can we deduce from the observation of phenomenal experience some of the reasons why it does exist as a biological endowment of our species? Or can we consistently accept the claim that phenomenal experience be a biological epiphenomenon? The usual way to understand why a given biological feature was selected in the phylogenetic past is to consider its function, and to examine how this function could be advantageous for those individuals endowed with it. Unfortunately, phenomenal consciousness seems to fulfill no obvious function. The fact that it plays a definite role in perceptive processes or in the production of behaviour, in itself, is questionable. Furthermore, what such a role would consist in remains mysterious, so that it seems pointless to speak of its adaptive value. 5. Several authors mention strong arguments supporting the idea that conscious experience may be an epiphenomenon [START_REF] Jackson | Epiphenomenam qualia[END_REF][START_REF] Flanagan | Consciousness Reconsidered[END_REF][START_REF] Flanagan | Zombies and the function of consciousness[END_REF]. It has often been claimed that our current knowledge of the laws of nature did not allocate any definite role to consciousness: "the view that qualia are epiphenomenal is a perfectly possible one" [START_REF] Jackson | Epiphenomenam qualia[END_REF]. Flanagan demonstrates that "conscious inessentialism" is consistent: "[Conscious inessentialism is] the view that for any intelligent activity i, performed in any cognitive domain d, even if we do i with conscious accompaniments, i can in principle be done without these conscious accompaniments" [START_REF] Flanagan | Consciousness Reconsidered[END_REF] 6. Since there is no absolute way of telling the difference between sentient and non-sentient beings from the observation of behavioural performance, the epiphenomenality of qualitative experience cannot be ruled out, and we are left with an evolutionary puzzle:

"Telling a convincing story about the adaptive advantage of consciousness is very hard. The upshot is that there exist no good stories for why consciousness evolved in this actual world. There are as yet no credible stories about why subjects of experience emerged, why they might have won or should have been expected to win  an evolutionary battle against very intelligent zombie-like information sensitive organisms." [START_REF] Flanagan | Zombies and the function of consciousness[END_REF] _______ 3 Some authors consider that presupposing the existence of qualia is already a faulty attitude (Dennett 1988). We will come back to this issue in the discussion, section 9.

7. If we can imagine that fully functional creatures may exist and behave as we do without having any conscious experiences at all, how can any serious claim be made about the biological role of phenomenal consciousness? Qualia, these qualitative aspects of experience, the "What it is like" character of mental states [START_REF] Nagel | What is it like to be a bat ?[END_REF], would appear as mere superfluous and fortuitous adjuncts to brain activity. Even if conscious inessentialism does not rule out a possible role for qualia, it makes epiphenomenalism appear as a perfectly sound doctrine. 8. Most of the complex processes going on in our body are indeed achieved without involving any conscious component. We are not conscious of our immune system, we do not feel each contraction of our stomach, we are not aware of maintaining our equilibrium at each moment. Many of our cerebral processes are performed without us being aware of them. The way we analyse a visual scene, the way we recognize words in a complex acoustic signal, the way we adapt our walk on an uneven ground are good examples of such processes. If we think that complex processes like shape and object recognition are performed unconsciously, we may wonder why perception involves consciousness at all. Why are we sentient beings, why are we not unconscious, robot-like creatures? What use is the ability we have to experience internal mental states or events in the outer world? 9. From a Darwinian perspective, the issue becomes even more enigmatic: if phenomenal consciousness plays no causal role in neural processes, how and why was it retained by natural selection? 10.

The question of the very existence of qualia is one to the most difficult issues addressed by contemporary philosophy and cognitive sciences. It is a form of mind-body problem. Understanding the reason why we are feeling anything at all belongs to the so-called "hard problem" [START_REF] Chalmers | Facing up the problem of consciousness[END_REF]. Such questions have been discussed at length by many authoritative authors. What is proposed in this paper is to offer the external view of an engineer on one limited aspect of the problem, the epiphenomenality of qualia.

11.

In what follows, I will first consider the possibility that qualitative aspects of experience are evolutionary epiphenomena. Then, in order to evaluate such a hypothesis, I will consider the way qualia are structured within a given sensory modality and their relation to sensory inputs. I will give evidence suggesting that this structure is, from an engineering perspective, locally "optimal". The next step will be to claim that such local optimality is not here by chance, but must be ascribed to natural selection. The conclusion will be that qualia epiphenomenality should be considered unlikely: if there was a selective feed-back that enabled the emergence of locally optimal qualia, these qualia must be efficient in some way. The argument is rather indirect, though, and the questions of how mental qualities can be efficient and even why they exist at all will remain out of the scope of this development.

Epiphenomena and evolution

12.

Evolutionary epiphenomena are known in biology. Features which were not selected for, but result from the selection of other characteristics, are evolutionary epiphenomena. A trivial example is the colour of our internal organs, which is a mere property of organ tissues, but which was certainly not selected for its aesthetic value. Such epiphenomenal characteristics have been named "spandrels"4 .

13.

Any organ or behavioural disposition that is recognized as the direct product of natural selection can be shown to be locally "optimal". For example, any minor change in eye architecture would result in a less efficient device for its typical owner. Such optimality is local. Major changes that would lead to better performance are not offered to natural selection. For instance, evolutionary biologists often mention the fact that the neural wiring of photosensitive cells would be better located behind the retina, as it is in mollusk eyes, rather than in front of it, as is the case in vertebrate eyes. But such a new design cannot be reached within a small mutational step, and intermediary mutants would have a very poor vision indeed. Gould and Lewontin rightfully denounce "panglossism", the doctrine of Voltaire"s character Pangloss for whom every feature of the world is optimal [START_REF] Gould | The spandrels of San Marco and the Panglossian program: a critique of the adaptationist program[END_REF]. Global optimality is not a sound concept of biology. Natural selection can only achieve local optimality. The vertebrate eye is not the best possible organ for vision, but it is locally optimal, and this local optimality can be directly assessed.

14.

A common misunderstanding about evolution is that it is a slow process. As a consequence of this supposed slowness, all biological forms would be on the way of their improvement, and none of them would, even locally, be optimal. The fallacy comes from the fact that the judgement is made at the wrong time-scale. The mechanisms of evolution that make it rapid (as illustrated by bacterial adaptation to antibiotics) have been better understood since they are reproduced by engineers in an optimisation method called Genetic Algorithms [START_REF] Holland | Adaptation in natural and artificial systems[END_REF][START_REF] Dessalles | L'ordinateur génétique[END_REF]. Genetic recombination and selection give rise to what has been called intrinsic (or implicit) parallelism [START_REF] Holland | Adaptation in natural and artificial systems[END_REF]. Evolutionary change in presence of selection pressure operates in parallel in the population and rapidly reaches equilibrium, characterised by a local optimum [START_REF] Dessalles | L'ordinateur génétique[END_REF]. Subsequent changes may be slow: they suppose a transition from the current local optimum to another, giving rise to a series of punctuated equilibriums [START_REF] Gould | Punctuated equilibria : the tempo and mode of evolution reconsidered[END_REF]. We must retain that any sub-optimal biological set of features is rapidly improved until it reaches a local optimum.

15.

Gould and Lewontin also denounce pan-selectionism, a doctrine claiming that all features of living beings have been selected as such. There are important exceptions: neutral characteristics and spandrels. For obvious reasons, neutral characteristics lack local optimality. Spandrels do have some local evolutionary necessity. For instance, the colour of our internal organs is, in some sense, locally necessary. Any change that would produce a new coloration would result from a change in the histological structure of organs. If, as we may suppose, this structure is the result of natural selection, such a modification will lead to suboptimal organs. The local evolutionary necessity of a spandrel thus results from the local evolutionary necessity of other attributes, to which it owes its existence. In other words, the local evolutionary necessity of a spandrel is inherited from what causes it, whereas the local evolutionary necessity of a functional feature is due to its indirect effects on survival (or, ultimately, on gene propagation). _______ 16.

The crucial difference between spandrels and functional features is that the local evolutionary necessity of spandrels in only indirect. Since they do not have reliable effects on survival, their structure cannot show any optimality per se 5 . The green colour of bile is not locally necessary per se. We cannot account for it unless we consider the function of bile itself and the local optimality of its chemical composition.

17.

Similarly, if we consider that consciousness is a mere epiphenomenal property accompanying some neural mechanisms, it must be an evolutionary spandrel, a fortuitous byproduct of brain phylogenetic evolution. As Jackson puts it: "The point is that all we can extract from Darwin"s theory is that we should expect any evolved characteristic to be either conducive to survival or a by-product of one that is so conducive. The epiphenomenalist holds that qualia fall into the latter category. They are a by-product of certain brain processes that are highly conducive to survival." [START_REF] Jackson | Epiphenomenam qualia[END_REF] 18.

As such, qualia should not show any feature that could be directly identified as optimal. I will now suggest that this is not the case, and that qualia do exhibit an optimal arrangement. As Nichols and Grantham put it, "the phenomenal consciousness system exhibits a level of anatomical complexity that plausibly requires an adaptation explanation" [START_REF] Nichols | Adaptive complexity and phenomenal consciousness[END_REF].

The structure of qualitative spaces

19.

For each sensory modality, there is a qualitative space: qualitative experiences are more or less similar, depending on input stimuli. As I will suggest, this fact is highly unexpected from an epiphenomenalist perspective. Consider for example colour perception. The organization of colour experience for a given subject is well represented in a threedimensional space [START_REF] Munsell | A color notation[END_REF][START_REF] Clark | Sensory qualities[END_REF]. This fact, in itself, is opaque to introspection6 . It can be proved using psychological techniques. This finding is not deduced from physiological considerations, like the fact that we have three types of photosensitive cells in the retina7 . The structure of qualitative colour space can be inferred from subjective judgments alone. From a set of similarity judgments made during controlled experiments, like "this colour is closer to this one than to that one", a colour map can be deduced using _______ 5 To show how the local evolutionary optimality of a functional feature and a spandrel differ, consider a characteristic E, for instance the aerodynamic efficiency of the wing of a bird, as a function of several relevant parameters a i (e.g. the length of the wing, its curvature, etc.) An aerodynamicist could verify that actual birds show values {a i * } that maximize E: E/a i (a 1 * , a 2 * ,…) = 0. Now, consider a spandrel s (e.g. the distance between the end of the wing and a red spot on the neck of the bird). We can write s = S(a 1 ,b) where the parameter b determines the position of the spot. The value of s at equilibrium is constrained: s * = S(a 1 * ,b * ), where b * is determined by independent evolutionary constraints on b. In this sense, s has, indirectly, a local evolutionary necessity. However, s * is not itself the solution of the aerodynamic problem. If the flight efficiency is rewritten as E 1 (s,a 1 ,a 2 ,…), then no definite value s * corresponding to the maximum of E 1 can be determined, neither by the aerodynamicist not by natural selection. As b may vary across generations without altering flight efficiency, s may vary without affecting E 1 . We thus have E 1 /s = 0 for any s. The spandrel s is not a reliable parameter of flight efficiency.

techniques like multidimensional scaling8 . Such experiments give results that are compatible with a three-dimensional colour space [START_REF] Shepard | The analysis of proximities: Multidimensional scaling with an unknown distance function[END_REF][START_REF] Clark | Sensory qualities[END_REF]. The distribution of colours in this space can be represented as a solid shape. Intensity is one axis of this solid. For a given intensity value, we get a disk-like "slice" of the solid. Colours corresponding to this intensity value are distributed over the disk: gray near to the center and saturated colours at the periphery, arranged in the same order as in the rainbow.

20.

There is more to be said about this colour space. In each disk-shaped slice, which corresponds to a given intensity value, the blue-yellow axis and the green-red axis constitute two roughly orthogonal diameters9 . Moreover, not all colours are equivalent. For each subject, there seems to be four "unique" colours, located in the green, yellow, blue and red areas. All other hues are experienced as mixtures of two neighbouring unique hues. There is, however, a significant variance among subjects about unique colours that leads some authors to question the scientific value of the concept [START_REF] Saunders | Are there non-trivial constraints on colour categorization?[END_REF].

21.

The existence of a subjective space for colour perception, characterized by a 3Dstructure, is not an exception. Other modalities also give rise to qualitative spaces. Qualitative taste can be approximated by a tetrahedral space, with four basic tastes at the vertices [START_REF] Clark | Sensory qualities[END_REF]. Sound intensity and pitch are distributed along one-dimensional spaces, whereas judgements about timbre require four dimensions to be only approximated [START_REF] Bonnet | Traité de psychologie cognitive[END_REF]. In what follows, I suggest that this organization of qualia in definite spaces is not an anecdotal curiosity, but must result from a specific requirement.

Why are there qualitative spaces ? An engineering hypothesis

22.

The existence of qualitative spaces, although their global structure is opaque to introspection, is not at odds with intuition. Colours, smells, sounds can be compared from one stimulus to the next. This statement is obvious, so obvious that we fail to notice that it could be otherwise. If qualia were genuine epiphenomena, it should be possible that they form no identifiable structure. It would even be possible that qualia in a given sensory modality cannot be compared. This would be a logical extension of what thought experiments about inverted spectra suggest. According to such "experiments", qualitative experience may be distorted from one individual to the next without producing any behavioural difference: "Though you and I have exactly the same functional organization, the sensation that you have when you look at red things is phenomenally the same as the sensation that I have when I look at green things. If this hypothesis is true, then there is a mental state of you that is functionally identical to a mental state of me, even though the two states are qualitatively or phenomenally different. So the functional characterizations of mental states fail to capture their "qualitative" aspect" (N. [START_REF] Block | Are absent qualia impossible?[END_REF] 23.

What such experiments show is that there is no way to prove that, when looking at a cherry, a person A has the same qualitative experience as another person B. To go beyond the _______ usual assumptions of such experiments, which merely postulate simple permutations or global inversions of qualia, why not suppose that colours experienced by A correspond to a complex permutation of B"s subjective colours, or that A and B"s experiences are radically different ? Such a discrepancy in personal experiences would remain unnoticed, since both A and B would use the word "red" to qualify cherries. If we believe that qualia can be manipulated this way without any predictable consequence, however, then the very existence of finely structured qualitative spaces becomes a wonder. Conversely, if we consider the structure of qualitative spaces, as it can be deduced from psychological experiments, from an engineering perspective, we must conclude that most qualia permutations, during thought experiments, are not admissible. Moreover, the structure of qualitative spaces makes sense if these spaces are part of a definite functional design. Several observations will lead us to these conclusions.

Continuous projection

24.

The most obvious feature of qualitative spaces is that the stimulus-quale projection is continuous. Two close visible wavelengths will be experienced as similar hues. In our perception of the rainbow 10 , there is a smooth transition between colours. We could think of another picture: the rainbow, which results from a continuously varying physical parameter, could be experienced as kind of jumble: similar colours would occur at different locations ; neighbouring colours would be experienced as strongly dissimilar ; there would be no order consistent with the topology of our qualitative space. In our actual experience of rainbows, though, yellow and red are separated by all distinguishable orange hues, the space between blue and green is like greenish blue and bluish green. By independent means, for instance technical devices, we are able to determine a stimulus space structure and to control the stimulus presented to subjects. When the stimulus is modified in a continuous way, as in the rainbow, subjects" experiences vary continuously.

25.

This observation is a genuine psychological result. Qualitative spaces, in each modality, are structured in a definite way, as shown in great detail by Clark. Stimulus spaces also have their own structure. The fact that both structures correspond through a continuous relation is not a trivial phenomenon. In some cases perception, e.g. phonological perception, is consciously experienced as discontinuous [START_REF] Harnad | Psychophysical and cognitive aspects of categorical perception: A critical overview[END_REF]), but the amplitude of discontinuities remains small 11 .

26.

The same seems to hold for all sensory modalities, even if it is not equally easy to demonstrate. We expect that molecules with similar chemical structures are likely to project onto neighbour points of taste qualitative space, as anyone can experience with various sugars (monosaccharides and disaccharides) or various types of salt. Continuous projections between stimuli and subjective experience are more apparent for sound intensity, sound pitch, warmth or pressure exerted on skin. For instance, sound intensity subjective perception can be approximated by a continuous function of acoustic pressure [START_REF] Bonnet | Traité de psychologie cognitive[END_REF]). The existence of continuous stimulus-quale projections is a first fact that restricts what subjective experience can be like. _______ 10 Actual rainbows offer less than seven discernible hues. We use the term rainbow to refer to a complete set of monochromatic stimuli at a given intensity. 11 If p represents the stimulus-quale projection, then p can be considered continuous. In the relaxed version, p must be such as |p(s+h)p(s)| < C for any stimulus s. The constant C measures the degree of continuity, and h measures the granularity at which qualia are considered

Directionality

27.

The stimulus-quale projection has another important property: it may be directional. This means that for some situations like the rainbow, qualia are not duplicated and appear to be totally ordered: we do not experience a U-turn when the stimulus increases. In other terms, the stimulus-quale projection is monotonic 12 . When we look at a rainbow, the colours seem well ordered and are not repeated. When the wavelength increases, we do not experience yellow twice. The same seems to hold for other modalities. Increased pressure or increased heat transfer on our skin generate new experiences 13 .

Regularity

28.

The projection of stimulus space on qualitative space in a given sensory modality may have another important property. When we look at a rainbow, we experience several colours and a smooth transition between them. Moreover, these colours are gracefully distributed along the spectrum. This, also, is an unexpected phenomenon. We could easily imagine that one colour, some pure red for instance, occupies 90% of the visible spectrum, while all other hues are concentrated on a 10% of the available wavelength range. Colours seem, instead, to share the subjective colour spectrum in such a way that they are, more or less, evenly spread out over it 14 .

Dynamic range

29.

Qualitative spaces offer a variety of vivid and rich experiences. Though it may be difficult to quantify precisely, the dynamic range can be defined intrinsically as the ratio between the greatest difference in the qualitative space and the smallest discernible difference between two qualia. In the case of colour space, we may try to estimate the dynamic range by counting the maximum number of discernible hues between two colours. Some 150 colours can be discerned in monochromatic waves; other dimensions of colour offer smaller dynamic range: one can distinguish some 25 saturation levels in the blue, and some 30 intensity levels between black an white [START_REF] Sève | Physique de la couleur[END_REF]. Different qualitative spaces may differ significantly in their dynamic range. Sound intensity, for example, is perceived on a subjective scale with a dynamic range of 2000, while pitch is experienced with a maximum dynamic range of 3000 _______ 12 For some one-dimensional stimulus sets (e.g. pure wavelengths at a given intensity), corresponding experiences are structured as an ordered, one-dimensional space as well. Moreover, for any two non-equal stimuli s 0 and s 1 , the sign of (p(s 1 )p(s 0 ))/(s 1s 0 ) does not change. If the stimulus-quale projection p can be considered as derivable, then directionality is captured by the condition that p'(s) does not change sign, where s stands for the current value of the stimulus. 13 Directionality does not prevent the qualitative space to loop on itself. The last colour of the rainbow is close to the first one; some individuals have a brief experience of very high water as if it were cold (before pain is felt). For such situations, we may consider the relevant qualitative space as periodic, i.e. for any quale q, q+A = q, where A stands for the total range of the relevant qualia, and consider directionality as a local requirement, namely that (p(s+h)p(s))/h does not change sign for some granularity h. 14 The requirement that qualia are evenly distributed over a given stimulus range can be captured, in the case of a one-dimension stimulus space, by a condition on the first derivative of the stimulus-quale projection: |p'(s)| < K, where the constant K measures the regularity of the distribution. This condition prevents experience to change too rapidly when stimuli changes are small. If p cannot be considered as derivable, the condition must be replaced by |(p(s+h)p(s)) / h| < K, where h is the granularity at which qualia are considered. [START_REF] Bonnet | Traité de psychologie cognitive[END_REF]. Notice that contrary to colour spaces, these ranges are quite far from uniform, loud sounds or high tones being much less distinguishable. Our point is that all qualitative spaces possess a significant dynamic range.

30.

These four properties: continuity, directionality, regularity and dynamic range are not, as I will suggest now, fortuitous features. In order to clarify this point, we must make a small digression.

Evidence from communication theory

31.

Communication theory considers optimal projections. In digital communications, each elementary event is the transmission of a symbol taken from an alphabet. This alphabet may be as small as {0,1}, but it may also contain a significant number of possible symbols (more than 1000 for a typical telephonic modem). Symbols are abstract mathematical objects, but when it comes to actual transmission, they must be converted into physical signals. Let us assume that such signals belong to a two-dimensional space: the receiver will for instance measure two tensions v 1 and v 2 on two independent wires 15 . If we want to transmit with a twosymbol alphabet (like the Morse code or any binary code), then the corresponding signals should be chosen on two opposite corners of the available space: for instance a null signal for 0, a maximal value on both wires for 1. If we want to transmit messages based on a foursymbol alphabet, then the signals corresponding to the symbols should be placed on the four corners of the available signal space. If the number of symbols is large, the corresponding signals will be located at the vertices of a hexagonal lattice, each signal being surrounded by six neighbours (figure 1).

32.

Why are such solutions optimal? The main problem in telecommunication is noise. Received signals are not identical to emitted signals. The problem is to avoid errors when recovering the abstract message from its physical realization. If signals representing symbols are maximally separated from each other, then the effect of noise on error rate is minimal. Hence the hexagonal pattern when many points have to be optimally spread out on a 2Dspace. I will suggest that the structure of qualitative spaces is constrained by a similar requirement.

33.

Qualitative spaces possess an intrinsic topology, however. They do not consist in a set of non-related points. This may also be the case in digital communication, when there is a topology in the set of symbols, i.e. when symbols are more or less similar to each other. The consequence of errors may be significantly different, depending on whether the erroneous _______ 15 A more sensible example would be two parameters of a modulated carrier wave, as in quadratic amplitude modulation.

v 1 v 2
symbol is close to the correct one or far from it. If we consider the coding of music, for instance, a small shift in the signal space should not lead to distant values of the acoustic signal. If this constraint is respected, most actual errors will remain unnoticed. This has a consequence on the choice of the symbol distribution over the available signal space. An obvious solution is to place signals associated to neighboring symbols close to each other. As a consequence, the projection from signals to symbols is continuous. This is also a property found in the case of qualitative spaces.

34.

In order to make the analogy with the structure of qualitative spaces more concrete, imagine we have to design a technical device to detect visible wavelengths, using a given number of sensors {S 1 , S 2 , …S n } connected to a set of light-emitting diodes {d 1 , d 2 , …d n }. Let us assume that this latter set has an intrinsic topology: d 8 is closer to d 7 and d 9 than it is to d 1 or d 10 . For instance, S i may control the switching of the ith element of a row of light-emitting diodes (figure 2a). The best choice, when it comes to tuning the detecting elements S i , is to divide the spectrum into n equivalent parts and have each d i responding to the ith part of the spectrum. The solution results from a single requirement: keep the average effect of errors as small as possible. This requirement leads to four constraints: (1) continuous (topology preserving) projection; (2) connex receptor fields; (3) regular distribution; and (4) significant dynamic range. Thanks to (1), the topology of the diode row can be used to keep errors harmless. When, because of a small error, d 7 goes on instead of d 8 , the consequences may be almost ignored, considering that d 7 and d 8 are neighbours. If the projection were discontinuous (figure 2b), there would be no way to distinguish between small and large errors. Since the former are by far more frequent, a continuous projection, by reducing their effect, is much preferable 16 . The role of ( 2) is, very basically, to avoid indeterminacy. If d 4 can be triggered by two non-neighbour sensors (figure 2c), the performance of the system is affected by a fixed error rate. The main effect of (3) is to lower error probability. If S 7 responded to a major part of the spectral range while other detecting elements were crammed on the remaining part (figure 2d), then it becomes much more likely that noise affecting one of the crowded sensors will cause an error. The effect is rather dramatic: starting from an error probability of 10 -3 when sensors are regularly spaced, we get an error probability as high as 0.38 when all but one of the sensors share 10% of the available range17 . The effect of (4) is to lower the effect of quantification (figure 2e). A finite number of sensors on a continuum introduce a quantification noise, due to the imprecision of readings on the diodes. If the dynamic range is divided by 2, the quantification noise is multiplied by 4 or, equivalently, is increased by 12 dB18 . Notice that the effect of (3) is also to avoid the unnecessary quantification noise due to the imprecision of S 7 (figure 2d).

_______

[a] [b] [c] [d] [e]

35.

Our analogy allows us to understand the role of the four properties of the stimulusquale projection emphasized above: continuity, directionality, regularity and dynamic range. All four properties make perfect sense if they are involved in a discrimination process. The role of continuity (e.g. the fact that colours follow each other quite smoothly on the rainbow) is to limit the effect of errors; the effect of directionality, in one dimension (e.g. the fact that colours occur only once on the rainbow), is to ensure the connexity of the projection (the fact that two disconnected stimulus ranges do not project on the same quale, introducing a systematic error source). The role of regularity (e.g. the fact that no single colour occupies 90% of the rainbow) is to minimise error probability and quantification noise. High dynamic range (e.g. the number of discernible colours in the rainbow) keeps the quantification noise low.

36.

In the context of communication theory, such properties unavoidably suggest that qualia are involved in a discrimination process. Before drawing some consequences of this observation, we must make two remarks. First, we should not always expect strictly equal shares from the regularity of the stimulus-quale projection. If some parts of the stimulus space are more relevant than others, we expect more efficient discrimination in these zones to be correlated with a higher density of qualia 19 . If we follow the engineering guideline, what is important is that the practical consequences of errors are kept low. The correlation between stimulus relevance and quale variation should be, in principle, empirically testable.

37.

The second remark concerns the inevitable distortion introduced in the stimulus-quale projection. The light spectrum emitted by a punctual source belongs to a space of infinite dimension. Our perception projects it onto a three-dimensional colour space. Such a projection cannot be made without distortion. Hence the fact, called metamerism, that many different stimuli are experienced identically. For instance, a superposition of two wavelengths corresponding to red and green will be experienced as an orange hue, as would be a single, intermediary wavelength. What we called a regular distribution of detecting fields can be represented as an optimal "folding" of the target space when projected back into the stimulus space, sometimes called "snake in the box" by theoreticians of error correcting codes.

Constraints on the inverted spectrum experiment

38.

The fact that qualitative spaces may be triggered through a continuous, directional and regular projection, explain why their manipulation during inverted spectrum thought experiments is so delicate, sometimes believed to be practically impossible [START_REF] Shoemaker | Functionalism and qualia[END_REF], Clark 1996a[START_REF] Palmer | Color, Consciousness, and the Isomorphism Constraint[END_REF]. It is indeed difficult to depict a scenario where experiences of red and green are inverted but all structural properties between colour experiences remain identical. For instance, you can"t simply change a single quale. Suppose two persons A and B have the same colour experience, except for one quale, triggered when seeing a ripe cherry.

_______ 19 Suppose that a relevance pattern is available, i.e. the relevance (between 0 and 1) of each stimulus zone is given by R(s). R can be seen as a distortion of the stimulus space: dt = R(s) ds. The new stimulus t varies more rapidly in relevant zones, when R(s) is high. We have t = T(s), where T is the cumulated relevance:

      s s du u R s T 0
The projection p is expected to be regular on the distorted stimulus t:

K dt ds ds dp dt dp   
We get the new constraint: |p'(s)| < K R(s) for each s=T -1 (t), which limits the variation of qualia q = p(s) according to stimulus relevance (the lower the local relevance, the lower the quale variation).

Suppose that red A  red B , while C A = C B for any other colour name C (colour names associated with suffixes stand for qualia). Then the stimulus-quale projection cannot be continuous for both subjects, and at least one of them should notice it. For instance, when a single wavelength stimulus goes from, say, 625 nm to 675 nm (orange to red), one of the subjects will experience an abrupt transition that normal subjects do not experience. This effect of quale inversion limits the scope of the claim that qualia of different subjects may be changed at will in thought experiments with no observable consequence.

39.

We should not yet conclude that all human beings must have identical qualitative experiences when presented with the same stimuli. There are indeed several ways to save part of the inverted spectrum claim. We could imagine that while A and B share part of their colour experience, red B is a quale unknown to A, and that all intermediary colours, for instance between yellow B and red B , are also completely different from what A has ever experienced. I do not see how such a claim can be refuted. However, it requires half of the qualitative space to be transformed into a brand new qualitative dimension. There is little ground to suppose that two roughly similar brains could generate completely different sets of qualia with similar structural properties.

40.

A more plausible claim would be that A and B may have globally identical qualia sets, but that the stimulus-quale projection, though continuous for both subjects, is nevertheless different. We must keep in mind that the two projections must respect continuity, directionality and regularity requirements, which can be experimentally checked and verified for each subject. As a consequence, the hypothesised mapping from A"s qualitative space to B"s cannot be a jumble. It must be continuous 20 , directional 21 and regular 22 . If we deal with the same set of qualia for A and B, only approximations of global shifts or global inversions _______ 20 Technically, if we note p A the projection from the stimulus space on A"s qualitative space, and M the hypothesised mapping between A"s and B"s qualitative spaces, then p B = M o p A . Since p A and p B are continuous, M must be continuous too, at least on all qualia in A"s qualitative space reached by p A . 21 Using the notations of preceding notes, the directionality of M results from the directionality of p A and p B . Let"s consider two qualia q 0 and q 1 in A"s qualitative space. We suppose q 0 < q 1 . The corresponding qualia in B"s qualitative space are consistently ordered (for the sake of simplicity, we ignore the case of periodic qualitative space and consider that both projections are derivable):

              ds ' ' o ' ' 1 0 1 0 1 0 0 1 0 1          s s B B B s s A A q q s p s p s p ds p s p M dq q M q M q M
Here, s 0 and s 1 are some stimuli that trigger experiences q 0 and q 1 respectively for A. Thanks to the directionality of p A , the sign of (s 1s 0 ) is the same as the (constant) sign of p A '. If p B ' is positive, then (p B (s 1 )p B (s 0 )) has the sign of (s 1s 0 ), the opposite otherwise. We thus see that M is directional, and that its directionality equals the sign of p A '•p B '. 22 We can check the regularity of M by computing M' on any quale q = p A (s) experienced by A when exposed to stimulus s. Since p B = Mop A , we have p' B (s) = M'op A (s) p' A (s). This relation puts a constraint on M':

| M'op A (s)||p' A (s)| < K B where K B measures the regularity of p B . The only possibility for M' to reach high values is when p' A (s) is small or zero. At such locations, B would experience (rapid) qualitative change, while A would notice little change or no change at all. This discrepancy would contradict the basic assumption of the thought experiment that A and B would fail to communicate any difference in their experience. As a consequence, we must conclude that the mapping M between A"s and B"s qualitative spaces must be regular. are admissible. This constitutes already a significant constraint on colour spectrum manipulation23 .

41.

The preceding discussion does not rule out the theoretical possibility of spectrum discrepancies between two human beings. However, it highlights the importance of the continuous and evenly distributed stimulus-quale projection. Because of it, no change of the spectrum can be envisaged except global shifts or global inversions. The fact that actual qualia offer resistance to manipulation during thought experiments indicates that they are not as fortuitous as would be expected from epiphenomena. Qualitative spaces are organised entities, and their structure shows characteristics which appear optimally designed for discrimination. Now we may try to relate this observation to the fact that qualia played a role in our evolutionary past.

Qualia and evolution: an analogy

42.

Phenomenal consciousness is a biological characteristic of our species, and possibly of many others24 . From the preceding development, we conclude that qualitative spaces possess certain optimal properties. Is it sufficient to conclude that qualia play a causal role in cognition? It can still be argued that qualia, though optimally organized, are a mere accessory feature accompanying neural processes, that they are an effect without being the cause of anything. I propose here an indirect argument in support of the opposite option. In order to do so, I will use an analogy.

43.

When ethologists study bird songs or calls, they are able to describe certain features as locally optimal. For instance, the acoustic spectrums of some song show features like large bandwidth and frequency modulation that seem correlated with an easy location of the singing animal and species-specific recognition [START_REF] Hauser | The evolution of communication[END_REF][START_REF] Naguib | Estimating the distance to a source of sound: mechanisms and adaptations for long-range communication[END_REF][START_REF] Mathevon | Sound-based species-specific recognition in the blackcap sylvia atricapilla shows high tolerance to signal modifications[END_REF]. Songs performing similar functions in different species tend to show the same type of characteristics. For instance, the maximum frequency of the song and intervals between notes are correlated with habitat density across species (van Buskirk 1997). From the ethologist"s point of view, such acoustic characteristics are well designed for their observed function. They are not here by chance. Their local optimality only makes sense if they were produced by natural selection, since natural selection is regarded as the only source of locally optimal design in biology25 [START_REF] Nichols | Adaptive complexity and phenomenal consciousness[END_REF].

44.

If we consider now the bird pharynx (syrinx) from a physiologist"s perspective, it may be possible to show that some of its characteristics are locally optimal to emit a specific song. For instance, the complex syrinx of passerines, especially oscines, can be correlated in some detail with the complexity of their song. However, if we ignore the song, those characteristics _______ of the syrinx will remain incomprehensible. The local optimality of the syrinx structure can only be explained by the optimality of the song"s acoustic properties.

45.

Ancestors of song birds were selected for their ability to sing. Should we consider that singing itself was selected, or rather that the syrinx was selected in order to allow territory signalling? Perhaps we should look at the neural circuits involved in singing and say they were also selected for territorial signalling purposes. And what did selection retain after all, if not the genetic changes that make the difference between song birds and their non-singing ancestors? From genes to neural circuits, syrinx and song, there is a long chain of embryological and functional events. Each of these events explained by its role in allowing the song. When ethologists study those birds, they do not exclude the song, they do not consider that the sole syrinx was shaped by evolution to perform territory signalling.

46.

In this analogy, the song represents qualia, and the syrinx may stand for their neural correlates. What seems obvious in the case of avian ethology should hold for some aspects of the study of consciousness. Because they exhibit an optimal constitution, qualitative spaces, like bird song, should not be excluded from evolutionary accounts, but should rather be seen as a genuine link in the causal chain linking genes to biological function.

47.

We can go one step further and suggest that qualia should be taken into account if we want to find some evolutionary justification for the corresponding brain states. Consider again the causal chain going from genes to biological function. Genes are not directly selected. They produce phenotypes, which undergo competition and selection. From genes to phenotypic characteristics, a long causal chain of structures is involved. The efficiency of the phenotype for gene transmission is what confers a Darwinian justification to the whole chain. Each link is explained in reference to the next link in the chain. At one end of the chain, genes are mere DNA sequences. They have no intrinsic feature that may justify their existence. To justify their presence in a given species, one must ultimately locate some advantageous phenotypic characteristic that they contribute to cause.

48.

Phenotype is properly defined as the set of individual characteristics upon which natural selection directly acts [START_REF] Dawkins | The Extended Phenotype -The Gene as the Unit of Selection[END_REF][START_REF] Dessalles | L'ordinateur génétique[END_REF]. Qualitative spaces, like bird song, as far as they exhibit a locally optimal constitution, are either phenotypic themselves or part of the genotype-to-phenotype chain. Ignoring them would result in a broken chain with no evolutionary account. There would be no point in trying to explain why the syrinx has its actual characteristics if it is studied as if it belonged to a mute bird species. Likewise, neuroscientists cannot avoid considering qualia if they want to find some evolutionary justification for the presence of their neural correlates. The reason for this, once again, is that qualitative spaces show complex design that an engineer will inevitably ascribe to a discrimination function.

49.

The comparison with ethology is no more than an analogy. It is of course thinkable, though extremely unlikely, that qualitative spaces evolved their orderly structure by pure chance, without having been selected for that. It is also thinkable that this structure, which seems so nicely designed for discrimination, fulfils another function. But those who support one of these two options are now in trouble, either being suspected to believe in miracles, or being requested to show how the structure of qualitative spaces relates in a locally optimal way to their supposed function.

Discussion: qualia and ghost spandrels

50.

The conclusion we arrived at, that qualitative spaces are part of the genotype-tophenotype chain, has further consequences. Like any biological characteristic showing functional design, qualitative spaces must have been shaped by natural selection. This can only be understood if qualia make a difference, if they helped our ancestors to be our ancestors. This conclusion is indirect evidence in support of a functional role of qualia. Even if such a functional role is difficult to conceive, it would be even harder to believe that qualia could be subject to natural selection without having any effect on cognitive processing.

51.

An alternative account that may restore the epiphenomenality of qualia comes from the idea that the structure of qualitative space would be a mere consequence of physiology. This idea is captured by Chalmers" principle of structural coherence (in this quotation, awareness stands for cognitive processes):

"It is this isomorphism between the structures of consciousness and awareness that constitutes the principle of structural coherence. This principle reflects the central fact that even though cognitive processes do not conceptually entail facts about conscious experience, consciousness and cognition do not float free of one another but cohere in an intimate way." [START_REF] Chalmers | Facing up the problem of consciousness[END_REF] 52.

The principle presupposes an isomorphism. We rejected the possibility that qualia be mere by-product of neural activity, because as such, they would be evolutionary spandrels. Yet spandrels, contrary to qualia, show no intrinsic optimality. Chalmers could insist that qualia constitute a special type of spandrel, some sort of ghost spandrel. If qualia were a kind of echo of neural activity, qualitative experience could be epiphenomenal and yet appear as optimally structured. It would inherit its structure from its neural correlates.

53.

Such a claim is hard to refute, since Chalmers" isomorphism must exist ex post: any difference between conscious experiences must be ascribed to differences between their neural correlates. The question is to know whether qualia were the target of natural selection or not. My argument in favour of qualia is that the structure shown by qualitative spaces is simple. This structure is, yet, optimally related to stimuli to allow discrimination. The set of corresponding brain states must have a much richer structure. No neuroscientist would accept the possibility that projecting the brain states corresponding to colour vision onto a three dimensional space can exhaust what can be said of these brain states. Why would Chalmers" isomorphism just pick out the few properties of brain states that are relevant to optimal discrimination26 ? A proficient neurologist could perhaps trace acoustic properties of bird songs back to the corresponding brain states. But acoustic frequency and modulation are manifest in sound waves, not in brain states which encode them. Not only do qualitative spaces manifest optimal structure for discrimination, but they show no other structure. This constitutes a strong argument against the fact that qualia could be a kind of ghost spandrel, a faithful but superfluous echo of neural activity.

54.

The alternative hypothesis, which consists in considering qualitative experience as a key link in the genotype-to-phenotype chain, explains why the structure of qualitative spaces is both poor and optimal for discrimination. Such a view, however, requires that qualitative experience play a definite biological role.

_______

A biological role for qualia

55.

The preceding development about the locally optimal structure of qualitative spaces and its putative connection with discrimination functions should make no sense from an eliminativist perspective. Some authors stress the fact that the notion of quale is very difficult to ground epistemologically. It is claimed that the notion of quale is vague and ill-defined, and that any phenomenon accounted for by a pure change in qualitative experience can also, in principle, be explained by physiological modifications (Dennett 1988). As any scientific concept, the notion of quale is as temporary as was the notion of light before it was reduced to electromagnetic phenomena. However, in the present state of knowledge, it seems difficult to describe the properties of experience that were highlighted here (continuity, directionality, regularity, dynamic range) without considering qualitative spaces. In the case of temperature, for instance, every scientist can check for herself those properties: higher heat transfers are experienced as warmer. Such simple facts are not predicted in eliminativist frameworks. They nevertheless constitute a genuine scientific problem that demands scientific explanation. One may object that qualia cannot be recorded and that verbal reports do not constitute an ultimate evidence of their existence. This is not sufficient to rule out qualia from the scientific scene. In conventional science, scientific observers are never sure to watch exactly the same phenomenon. They rely on the fact that their perceptions (possibly mediated by some artificial device) are correlated to conclude that they are dealing with a genuine scientific fact 27 . It is not absurd to rely on internal checking when it comes to phenomena like continuity, directionality, regularity or dynamic range of qualitative experience. The fact that we can correlate our observations through language and mathematical formalism is sufficient to proceed with standard scientific procedures.

56.

The central notion needed to establish the main claim of this paper is the notion of qualitative space. We do not need to consider other aspects of consciousness and their relation to phenomenal consciousness [START_REF] Block | On a confusion about a function of consciousness[END_REF][START_REF] Zalla | Unité et multiplicité de la conscience : une étude critique des théories contemporaines à la lumière d'une hypothèse modulariste[END_REF], though some authors may find it impossible to isolate phenomenal consciousness from higher aspects of cognition. For instance, crucial issues about qualia are the nature of their intentionality and of their mode of presentation, which may depend on language and/or a theory of mind [START_REF] Carruthers | The evolution of consciousness[END_REF] or not [START_REF] Siewert | The significance of consciousness[END_REF]. Strictly speaking, we do not have to decide about such matters. Our discussion is about the structure of qualitative spaces, not their relation to other aspects of cognition (like memory, theory of mind, concepts or attention). We should note, however, that our claim about the evolutionary role of qualitative spaces is hardly compatible with a position that would restrict phenomenal consciousness to humankind. Moreover, relevant properties of qualitative spaces (continuity, directionality, regularity, dynamic range) are not predicted by accounts that make them result from high-level cognitive abilities like language or theory of mind.

57.

Qualitative spaces like colour space, for which there is some evidence of continuous, directional and regular stimulus-quale projections, seem to be optimally designed for discrimination. If, as we must presume, they result from natural selection, the next natural step _______ 27 Two individuals never see exactly the same rainbow in the sky at the same time, since the observer has to stay on the rainbow axis, at a definite distance, so as to see the rainbow with a 42.52° aperture. Similarly, two different persons, or the same person at two different moments, cannot be sure to deal with the same phenomenon when considering falling objects. Only correlations and mathematical laws like the law of gravity make them confident that their perceptions are alike. in the reasoning is to suppose that qualia must play a causal role for this discrimination function. There is some evidence in support of this causal role. In colour vision, for instance, many different physical stimuli are experienced as identical (see above, end of section 5). Nevertheless, such stimuli cannot induce different behavioural responses. We are aware of every colour shade that we can behaviourally discriminate. As noticed by Chalmers, "every color distinction corresponds to some reportable information" [START_REF] Chalmers | Facing up the problem of consciousness[END_REF]. This would be hard to explain if phenomenal consciousness was not involved in the functional process of discrimination.

58.

If we follow this line of argument, we should conclude that we perform discriminations on the basis of phenomenal qualities. From this perspective, phenomenal consciousness is what led evolution in our lineage towards increasing discriminatory capacities. One may wonder how we can define what is being discriminated thanks to phenomenal experience. From the engineering perspective adopted here, we don"t need to consider this issue: qualia ought to be involved in a discrimination process. For the sake of the evolutionary argument, suffice it to say that improved discrimination is always better (at equal cost) and is selected, no matter what is discriminated which correlates with food, danger, mating opportunity, etc. We may add that, since many animals share neural organisation with us and perform the same kind of perceptual discrimination, we can hardly suppose that phenomenal qualities are a human privilege, based on the use of language, conceptual thought or theory of mind, as is sometimes claimed 28 [START_REF] Carruthers | The evolution of consciousness[END_REF].

59.

The hypothesis about the functional role of qualitative experience predicts our actual phenomenal variety. If phenomenal qualities were epiphenomenal spandrels, even ghost spandrels, there would be no reason why our perceptions would give rise to such a variety of phenomenal states. How could we explain the structure, the richness and the vividness of the qualitative repertoire if it was a fortuitous echo of neural activity? Conversely, if qualia are causally involved in the process of discrimination, then we understand not only their structural properties, but also their variety and their intensity, what we called the dynamic range of qualitative spaces.

60.

Conscious species were selected according to the discrimination abilities that consciousness, for some reason beyond our reach, made possible. As a consequence, a rich gamut of phenomenal qualities was selected in each modality. The role played by conscious experience in discrimination is consistent with the fact that qualitative experience has a modular structure that systematically mirrors the functional organization of perceptual systems. It explains why phenomenal consciousness meets the fundamental constraints of signal discrimination efficiency by keeping relevant qualitative properties scrupulously apart. If, as claimed here, conscious experience is a powerful means for brains to perform efficient discrimination, then we must admit that it was selected for its own sake as an essential component of the perceptual process.

61.

Phenomenal consciousness should thus be considered as a proper biological character, on which natural selection could act. Any increase in qualitative variety was likely to induce a _______ 28 Part of Carruthers" claim, namely that phenomenal qualities are systematically available to higher cognitive processes [START_REF] Carruthers | Language, Thought and Consciousness[END_REF], remains perfectly sound in our context. What is no longer acceptable from the perspective put forward in the present paper is that phenomenal consciousness could emerge as an evolutionary by-product of the ability to read one"s own mind [START_REF] Carruthers | The evolution of consciousness[END_REF], even if it might be the case for other aspects of consciousness.

better discrimination ability and, within certain limits 29 , a more probable survival of individuals. Even if it remains far from obvious, we must consider that qualitative experience should be seen as a driving element in the evolutionary process which produced enhanced discrimination in perception.

Conclusion

62.

The purpose of this paper was not to give a formal proof for the existence of qualia. There was no attempt either to explain why perception is accompanied by experience. By emphasizing the fact that qualitative spaces are not randomly arranged, we pursued only one goal: it should be much harder, now, to think that qualitative experience can be epiphenomenal.

63.

The structure of qualitative spaces, which makes sense if qualia are involved in a discriminative process, gives us only an indirect evidence of the functional role of phenomenal consciousness. We concluded that qualia are not spandrels, that they must have some functional role, but this does tell neither why they are there, nor why they should be functionally necessary. The very nature of phenomenal consciousness is not addressed by the arguments we developed here. The claim about the optimality of the stimulus-quale projection concerns only the structure of qualitative spaces. It does not tell much about the intrinsic properties of qualia. The way we experience heat or taste, for instance, and the fact that it differs from colour experience, lie out of the scope of the argument, which only concerns the relationships between qualia within a given modality.

64.

What we showed, however, is that qualia cannot be excluded from a naturalistic account of our cognition, or even of our brain. Their structural properties, optimally designed for discrimination, strongly suggest that they are genuine links in the Darwinian selection loop going from genes to phenotype and back to genes.

65.

At this point, we are in a situation in which qualia, because they are not spandrels, must be given a role in the functional organization of the mind. The structure of qualitative spaces reveals that they do have such a role, and that this role may be related to perceptual discrimination. However, we still lack the conceptual frame that could help us understand how this role is carried out. It looks as mysterious as Newton"s forces through which corpses can act on each other at a distance. If we ever understand how qualia help us discriminate, we will also understand why evolution, at least in our phylum, had to produce phenomenal consciousness in order to design beings endowed with a powerful discriminative ability.

Figure 1 :

 1 Figure 1: Examples of optimal arrangements of signals in the available signal space.

Figure 2 :

 2 Figure 2: Illustration of [a] a continuous (topology preserving) and regular projection, [b] a discontinuous projection, [c] a continuous, though non connex projection, [d] an irregular projection and [e] a low dynamic range projection from a set of sensors S i to a display consisting of a row of light-emitting diodes d i .

S

The term refers to a metaphor used by[START_REF] Gould | The spandrels of San Marco and the Panglossian program: a critique of the adaptationist program[END_REF]: "the spandrels of San Marco". Spandrels are the triangular arch structures that support the upper domed roof of the cathedral of San Marco in Venice. These structures, which are integrated in the global decoration design as if they had been part of it in the first place, happen to unavoidably result from the presence of other architectural structures.

When colour TV was introduced, people needed some explanation to believe that all the variety of colours could be generated using only three basic colour components.

There are actually four kinds of photosensitive cells, but the central part of retina contains mainly three types of cells, the so-called cones.

For instance, by applying multidimensional scaling on a table giving inter-city distances, one can reconstruct a map with the location of all towns. This map may be rotated or inverted, as if seen in a mirror.

It should be stressed that the colour space of a given subject is only approximately structured this way. What we call "disk" here refers to an approximately round shape. Some colour-blind individuals lack one or two dimensions in their perception of colour.

For n sensors, the error amplitude average, for a small noise, is multiplied by n/3 when we go from a continuous projection to a random one.

In presence of a gaussian noise, the error probability is approximated by 2Q(A/(n)), where A is the amplitude equally shared among sensors,  is the noise standard deviation, and Q() is a rapidly decreasing function representing the probability for a gaussian variable to be higher than .

The amplitude of the quantification noise, for uniformly distributed signals, amounts to A 2 /(4D 2 ), where A is the signal range and D is the dynamic range.

Some further constraints come from the existence of four "unique" colours for every subject. Any shift or inversion of the spectrum must globally preserve the set of unique colours.

This does not preclude the philosophical possibility that artificial devices could have conscious experience.

This argument is taken as granted by most evolutionary biologists. As Pinker and Bloom say in their efforts to bring language back into the scope of evolutionary science, "there are clear criteria for when selectionist and nonselectionist accounts should be invoked to explain some biological structure: complex design"[START_REF] Pinker | Natural language and natural selection[END_REF]).

If, as is claimed here, qualitative space are much poorer than corresponding brain activities, then Chalmers" term isomorphism is inappropriate, and should be replaced by homomorphism.


Jean-Louis Dessalles
email: dessalles@enst.fr

A Structural Model of Intuitive Probability

Though the ability of human beings to deal with probabilities has been put into question, the assessment of rarity is a crucial competence underlying much of human decision-making and is pervasive in spontaneous narrative behaviour. This paper proposes a new model of rarity and randomness assessment, designed to be cognitively plausible. Intuitive randomness is defined as a function of structural complexity. It is thus possible to assign probability to events without being obliged to consider the set of alternatives. The model is tested on Lottery sequences and compared with subjects" preferences.

Introduction: Cognition and Probability

There seems to be an irreducible divorce between mathematical probability and human abilities to estimate odds. Various studies in psychology have revealed spectacular biases that seem to prove that human beings do not have a genuine consistent competence in probability. For instance, the gambler"s fallacy induces individuals to avoid betting on tail when tossing a coin if tails have been drawn several times in a row, as if the coin was expected now to fall on face to restore balance [START_REF] Terrell | A test of the gambler's fallacy: Evidence from pari-mutuel games[END_REF]). Another error, on which much has been written, is known as the conjunction fallacy [START_REF] Tversky | Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment[END_REF]. It consists in considering more likely that Linda, an anti-war militant female student, be found a few years later to be (a) a bank teller and militant feminist, rather than (b) a bank teller (without further precision). Normally, the latter case should be considered more probable, as it includes the former. A third famous error, called base-rate fallacy, consists in inferring probability from immediate evidence without taking a priori rarity into account, as when someone is sure to be ill when a biological test with less that 1% error says so, though that person has 33% chances to be in good health if the illness concerns only 2% of the population [START_REF] Evans | Bias in Human Reasoning -Causes and consequences[END_REF]. Moreover, people are over-sensitive to data presented as concrete and close and tend to ignore frequency statements [START_REF] Nisbett | Human inference: strategies and shortcomings of social judgments[END_REF].

Though it can be argued that such errors are unlikely to show in everyday life and need artificial testing situations to be revealed, they still seem to dismiss any possibility that human cognition involve a true competence in probability.

The problem is not only on the cognitive side. By various aspects, classical mathematical probability theory is unable to provide an acceptable theoretical frame for the corresponding cognitive abilities. Let us mention two reasons for this.

First, there are several cognitive modes of access to probability that are artificially conflated into one single concept through axiomatics and measure theory. One is symmetry: we feel compelled to assign a probability of 1/6 to each side of the dice, just because the dice is symmetrical. Another one is atypicality: if we see a cat which is three standard deviations away in size from our prototype of cat, we consider it improbable. Yet another is provided by frequency estimates in time, in space or in any relevant dimension: if all the fires I witnessed were located kilometres away from my home, I infer a spatial density for such events that makes me consider a fire next block quite improbable. From a cognitive modelling perspective, it is not obvious that these modes of access to probability can be unified the way classical probability does it, i.e. using measure theory, distributions and basic axioms.

Second, the foundations of classical probability theory are problematic from a cognitive modelling perspective. Authors like Franck Ramsey and Bruno de Finetti attempted to ground probability into intuition. They independently designed a procedure to found the concept of subjective probability [START_REF] Gillies | Philosophical theories of probability[END_REF]. Their idea was to imagine an experiment, in which the subject is asked to bet on the event. The subject does not know in advance whether it is a bet in favour of the event or against it. The relative height of the bet happens to reflect the subject"s estimate of the probability. Moreover, such estimates are shown to satisfy Kolmogorov"s axioms. The problem is that all classical definitions of probability, including de Finetti and Ramsey"s subjective probability, require explicit knowledge of all alternatives. De Finetti"s provocative statement, "Probability does not exist", claims that probability is grounded in the observer"s ignorance, not in some objective knowledge [START_REF] De Finetti | Theory of probability -A critical introductory treatment[END_REF]. Though, the "subjective probability" framework still requires some objective reference. To assign a probability to a given event, one must know the complete set of alternatives. How can non-omniscient beings assign probability at all? Let us take an example. Suppose a reader notices an unexpected fact about the novel she is currently reading. All six paragraphs in the current page begin with the character "C". The fact may be perceived as unusual by that reader because, obviously, she considers that the page could or should have been otherwise. What is the set of alternatives there? Alternative capital letters, alternatives letters, alternative paragraphs, alternative pages, alternative novels? What does prevent the reader from being overwhelmed with irrelevant details when considering alternatives, such as paper irregularities, scattered tiny ink flecks and the like? After all, the actual page is unique, and thus highly improbable, due to many tiny details that can be discovered in it. Maybe in no other (2006). In: D. Fum, F. Del Missier & A. Stocco (Eds), Proceedings of the seventh International Conference on Cognitive Modeling. Trieste,IT: Edizioni Goliardiche,[86][87][88][89][90][91] http://www.dessalles.fr/papers/Dessalles_06020601.pdf printed page, in the history of humanity, did the four words "excited", "deep", "talked", "gone" appear one above the other on four adjacent lines. Though, this latter fact would appear exceedingly boring to most of us. And the reason lies in the cognitive perception of probability. The repeated character "C" at the beginning of all six paragraphs of the page appears more improbable than the vertical arrangement of four particular words. We must explain why.

In what follows, we start from Solomonoff"s definition of algorithmic complexity, and we adapt it to cognitive modelling purposes. Then we define unexpectedness and base a new definition of intuitive probability upon it. The prediction that human subjects will find unexpected structures improbable will be explored in a little exploratory experiment. Finally, we will discuss the generality of the notions introduced in this paper.

A Structural Approach To Probability

In the framework of algorithmic complexity developed by Ray Solomonoff, Andrey Kolmogorov and Gregory Chaitin, the probability of an object is defined in relation to the shortest available descriptions of that object. The epistemological shift is significant. Probability is no longer a measure of the subject"s ignorance about the object, as in classical definitions. On the contrary, it depends on the subject"s ability to describe the object. More precisely, the algorithmic probability P M (x) of an object x is given by [START_REF] Solomonoff | Complexity-based induction systems: Comparisons and convergence theorems[END_REF]:

       i s L M i x P 2  (1)
This definition is relative to a given Turing machine M, and L(s i ) is the length of the i th string that, given as input to that machine, outputs a string beginning with x.  is a normalisation coefficient that depends on M and on the size of x. This probability is well approximated by:

P M (x)  2 -K(x) (2)
where K(x) is the Kolmogorov complexity of x. 1The concept of algorithmic probability captures the odds that a given device will spontaneously produce object x ("spontaneously", in the case of a Turing machine, means: when given a random sequence as input). For this definition to be of any use in cognitive modelling, the machine M has to be replaced by some model of the human cognitive ability to analyse objects. In the spirit of the definition of algorithmic complexity, we may say that any realistic computational cognitive model would be appropriate, as the resulting definitions will only differ by the complexity of discrepancies between these models, which is expected to be small.

We choose Michael Leyton"s Generative Theory of Shape as reference model [START_REF] Leyton | A generative theory of shape[END_REF]. According to Leyton, individuals analyse the structure of objects by constructing it as a multi-layered group. At each stage of this construction, a fiber group is transferred through a transfer group to give a higher-level structure. A simplified version of this theory will allow us to compute de complexity of structures. Consider for instance the following numeric sequence: 10 20 30 40 50 60 70 This sequence does not appear as random to a human mind trained to read numbers. One possible generative description of this sequence is as follows:

1. start with one uninstantiated number:

-2. transfer it through translation:

- - - - - - -
3. the number is a duplicated digit:

--------------

instantiate the second digit:

-0 -0 -0 -0 -0 -0 -0

dissociate the transfer operator:

-0 -0 -0 -0 -0 -0 -0 6. instantiate the first digit:

10 20 30 40 50 60 70

Several generative stories of minimal complexity can be proposed for the same structure, mainly by changing the order of the different phases (e.g. step 2 could be postponed after step 4). Valid generative sequences maximise transfer and recoverability [START_REF] Leyton | A generative theory of shape[END_REF]). In step 5, the transfer operator cop which performed a mere translation in step 2 becomes a joint operator +1/cop, with one first component that increments its operand.

To compute the complexity C R of the resulting structure, we propose to use the general formula:

C R = C F + C T (3)
C F represents the complexity of the fiber group, while C T is the complexity of the transfer group. Moreover, the complexity of instantiating a number is given by:

C n = log 2 (n+1) (4)
It represents the minimal information in bits to locate the number by its rank in the usual order.2 Some numbers may have a lower complexity. For instance, the number 33333 results from the transfer of the digit 3, and its complexity is C 3 +C cop +C 5 rather than log(33334) (C cop designates the complexity of the copy operation, and C 5 is the complexity of stopping the copy after reaching 5 terms). In our previous example, each step adds complexity: C cop for step 2; C dup for the duplication in step 3; 0 for the instantiation of step 4;3 C dup + C +1 for splitting the transfer operator and instantiating the first component to the incrementing operator in step 5; and lastly C 1 =1 for step 6. We may consider that C dup = C cop and that C +n = C n . The overall complexity is thus: C R = 3C cop +2 (we ignore that the sequence is bounded to seven elements, otherwise C 7 must be added).

If we apply formula (2) with C R as a human estimate of K(x), we obtain a paradoxical result. Simple structures like 33333 turn out to be more probable than more "random" numbers such as, say, 28561, in flagrant contradiction with what human subjects consider to be the case (see below). The logic underlying definitions (1) and ( 2) is that simple structures are more likely to be produced by any constructive device than complex ones. This assumption does not capture the human perception of randomness and probability.

In studying apparent randomness, [START_REF] Kahneman | Subjective probability: A judgement of representativeness[END_REF] observed that subjects require random sequences to be irregular. They mention that apparent randomness "is a form of complexity of structure", as "random-appearing sequences are those whose verbal description is longest." This observation is much in accordance with the modern definition of randomness, which is equivalent to the absence of structure (Chaitin 2001). However, it is at odds with the above definition of algorithmic probability: a regular structure is simple to achieve for usual devices, and should be considered more probable. The problem comes from the fact that human beings consider probabilistic outcomes to be produced, not by simple deterministic devices, but by random, and thus complex, devices. Therefore, we are not interested in some objective probability of the existence of an object, but in the probability of its occurrence as output of a complex machine. This is what the notion of unexpectedness captures.

Unexpectedness

To account for some important aspects of subjective randomness, we introduce the notion of unexpectedness as follows.

U(x) = C exp (x) -C obs (x)

(5)

where C exp is the expected complexity of object x and C obs is its actual, observed complexity. Let us explain the concept with an example.

If one is given a five-digit number, one expects a complexity C exp = C cop +5log(10), as a typical number results from a generative process such as (i) copy an uninstantiated digit; (ii) instantiate the five digits independently, after the copy, by choosing among ten possibilities. This is probably the complexity that will be assigned to 28561 by most subjects. If subjects observe that the given number happens to be 33333, they are surprised. This comes from the fact that 33333 results from a simpler process, of complexity log(10)+C cop , as the instantiation now occurs before the copy. The amount of surprise is given by U = 4log(10). Such a surprise is generally accompanied by a feeling of improbability (Dessalles, to appear). For instance, children may become excited and feel compelled to signal the event when their parents" car reaches 33333 on the clock. To capture this phenomenon, we define subjective probability as follows.

p(x) = 2 -U(x)

(6)

In our example, we get p=10 -4 , which corresponds to the mathematical probability of observing a number with five identical digits. We propose that subjective probabilities defined by ( 6) provide an adequate picture of the way human individuals perceive probability. 4 We designed a small experiment to test the validity of this claim.

The Lottery Experiment

We performed a few tests on a small set of subjects as a pilot experiment to test our model. Subjects informally approached in a café were offered a free French Lottery bulletin. They just had to choose 2 among 14 possible combinations of six numbers ranging from 1 to 49 to try their luck. Among these combinations, ten were fixed and four were randomly generated anew for each subject. The resulting lists of combinations were randomly shuffled to avoid artefacts due to the order of presentation. Some of the combinations offered to all the subjects, such as [1 2 3 4 5 6], [10 11 12 44 45 46], [10 20 30 31 32 33], [34 35 36 37 38 39], were chosen for their low complexity. Some others were the output of a random generator.

Twenty-six subjects were tested. Though all combinations are obviously equally likely to win, all subjects showed strong avoidance of the two simplest ones, namely [1 2 3 4 5 6] and [34 35 36 37 38 39]. When asked, all subjects expressed their strong feeling that these combinations were virtually impossible. Typical comments were "If you play that one, it"s sure you won't win" or "It would be stupid to play that one". There is only 3.10 -4 probability that the 26 subjects would have consistently missed these two combinations if they had chosen randomly. This result is consistent with findings about the avoidance of highly regular structures by lottery players [START_REF] Savoie | Évaluation et modification de conceptions erronées au sujet des loteries[END_REF]. It is also consistent with our claim that subjects spontaneously estimate probability through a difference of structural complexity. When the actual structure is too simple, the probability is perceived as too low for the event to have any chance to occur. By contrast, when the structure is complex enough, as for [6 17 21 28 37 42], subjects have no problem choosing the combination.

Though this pilot study is by far insufficient to give reliable results, we tried to estimate how far its results complied with the predictions of formula (6). To do so, we wrote a small computer programme in Prolog to assess the complexity of combinations. The programme maintains a "short-term memory", implemented as a list of four items maximum, which contains the last numbers or operations that were processed. The only operations taken into account are +1 and +2 increments. To reproduce the principle of formula (3), elements that are present in short-term memory are considered "free". Two sequences like 3 4 and 3 4 5 6 7 are thus of same complexity, since the last processed number and the +1 operation are always on top of the short-term memory. The programme detects increments, assigns intrinsic complexity to digits, and processes tenths and units independently if no global increment was previously detected. Though various details of the programme are necessarily arbitrary, they are of no consequence on the overall ranking of structures. Table 1 gives the simplest structures of the sample and their complexity as computed by the programme. Figure 1 shows the distribution of the combinations chosen by subjects according to their complexity. Note the total absence of low complexity structures (under 7). Though the pattern is globally compatible with our expectations, we may be surprised by the high score of structures with complexity between 9 and 12. It is likely that a few subjects did not pay attention to the kind patterns that appear on table 1. Some among them did not attempt to maximize the intuitive probability, but applied various alternative strategies, such as choosing combinations that are related to their personal life. For instance, one of them declared he chose combination [10 20 30 31 32 33] just because the last number was the area code of his mate"s birth location. Another indicated that he chose [8 9 26 27 28 29] because of the two first digits correspond to his birthday.

Obviously, the test cannot properly function with individuals who are aware of the equiprobability of all combinations. When informally testing colleagues or engineering students, we invariably obtained explicit blind strategies ("just take the first two ones") or meta-strategies ("I choose [1 2 3 4 5 6] to maximize my gain expectancy, as nobody will play it"). By contrast, none of the 26 subjects of the sample seems to have applied such mathematically informed strategy. This preliminary study provides nothing more than a hint that formula (6) may give a relevant estimate of subjective probabilities. We are planning a new experiment that will correct the shortcomings of this one. In particular, subjects will be made aware of sequential patterns and other conspicuous structures by being shown the Lotto grids as they will be played, together with corresponding combinations in numeric form. Moreover, we plan to investigate the remarkable influence of mathematical instruction on the subjects" choices.

Discussion

The outcomes of this preliminary study cannot be considered as validated results, and our next experiments with subjects will require significant improvement. However, it demonstrates that the phenomenon is real, that individuals base their judgement concerning probability on various cognitive factors, which include structure. It is not absolute structure that is relevant, but unexpected structure, i.e. structure that is less complex than anticipated.

It can be objected that the definition of probability given by formula (6) has no reliable character, as it crucially depends on the observer. For instance, as number like 28561 will appear random to most subjects, while a few of them will consider it as very peculiar, because they recognise the fourth power of 13. What would be a drawback for mathematical theorising (for instance, the French Lottery administration should ignore the structure of drawings) proves fundamental for cognitive modelling. Structure is indeed the only way to understand why subjects are confident in assigning highly diverging values of probability in the absence of any objective reason to do so. It also explains how individuals are able to assign probability to events when the set of alternatives is not defined in advance.

The little experiment on the Lottery reveals what many would consider as a bias. Most individuals, especially if they had no training in mathematical probability, fail to recognise the fact that a combination like [1 2 3 4 5 6] has no less chances to be drawn than [6 17 21 28 37 42] when it comes to making a betting choice. Such "errors" seem to demonstrate that aspects of human cognition are maladaptive.

In (Dessalles, to appear), we show that one of the main occasions on which individuals attend to unexpected states of affairs is to signal them in conversation. For instance, in the middle of a discussion between four people, one participant interrupted the conversation to observe that her three companions were wearing purple shirts. The unexpected decrease of complexity, as the set of purple shirts formed a simpler structure than usually observed within such a group, urged the young woman to make her conversational move. It is not structure per se, but rather the fact that she could observe an unexpected transfer (in the Leyton sense), that prompted her utterance.

A possible explanation for this sensitivity to structural complexity drop is that it may be advantageous in normal life, as unexpected structure is quite often correlated to hidden causality. Situations like National Lottery, in which honesty is guaranteed by law, are unlikely to occur in spontaneous settings. If [1 2 3 4 5 6] had been drawn on the very first day when National Lottery was created, most people, including mathematicians, would have been intrigued and would have suspected some hidden causality such as malfunction or malpractice, even if that drawing was as probable as any other, and no less probable than the actual drawing that elicited no suspicious reaction.

More generally, we may wonder to what extent unexpectedness, as defined by ( 5), is a relevant base for defining subjective probability. One of difficulty in modelling probability is to account for the fact that the same sense of improbability accompanies apparently unrelated phenomena, such as unexpected remarkable structures, atypicality and unexpected proximity. Formula (6) may provide a suitable answer. If we take into account the complexity of individuation of situations, then formula (6) offers a unified account of what people would consider improbable. A Lottery drawing like [1 2 3 4 5 6], which is remarkable by its unexpected low structural complexity, is easy to individuate within the set of usual drawings; 5 a one metre tall dog is much easier to individuate that a standard dog; the exact location of a fire that occurred in the immediate vicinity is easier to individuate than the location of remembered fires, which occurred tenths of kilometres away. In each case, the complexity of individuation is measurable. It corresponds to the length of the shortest ideal description that allows distinguishing the situation from any other possible situation. When this complexity is lower than usual, a feeling of improbability ensues, and its amplitude is predicted by formula (6). The same account in terms of complexity drop thus obtains for the various cognitive sources of improbability, while they were merely artificially merged in classical theories.

Perspectives

As already mentioned, much additional work is needed to validate the model by observing the way individuals behave when dealing with randomness and improbability. On the computer modelling side, we are currently working on algorithms that would offer a more faithful computer implementation of Leyton"s principles. The difficulty lies in the programme"s ability to detect operators that, when applied, diminish the complexity of the actual structure. Moreover, those operators must be cognitively plausible. The difficulty is that some operations may be easily detected by the computer and not by human subjects, and conversely. For instance, it is not obvious for subjects that the sequence 7 13 18 24 31 39 is remarkable because the gap between numbers is incremented; on the other hand, the mirror structure of a sequence like 2 14 29 35 35 29 14 2 is conspicuous to most subjects, but is not obvious for the computer to detect if it was not 5 It is a general result of algorithmic complexity theory that the number of simple structures of given size tends to be negligible in comparison to complex structures of the same size when that size increases [START_REF] Li | An introduction to Kolmogorov complexity and its applications[END_REF].

specifically programmed to do it. The sequence possesses Leytonian structure, as the pattern 2 14 29 35 is transferred through the symmetry group. The problem is that if the programme is given a variety of operators, we may face a combinatorial explosion when searching for existing transfers in the input structure.

Grounding subjective probability in unexpectedness offers new perspectives on probabilistic reasoning. It obviously provides a new and quantifiable interpretation of some probabilistic biases that were attributed to representativeness [START_REF] Kahneman | Subjective probability: A judgement of representativeness[END_REF]. It also accounts for the fact that human beings can assign low probability to unique events, without considering the set of alternatives, and it makes non trivial predictions on the influence of parameters like distance or recency on newsworthiness (Dessalles, in preparation).

The scope of the notion of unexpectedness, as it has been defined here as a difference of cognitive complexity, goes beyond the sole definition of intuitive probability. Unexpectedness is involved in what makes the interest of conversational narratives (Dessalles, in preparation), and in the definition of emergence [START_REF] Dessalles | Emergence in multiagent systems[END_REF]. Cognitive complexity, understood by analogy with Kolmogorov"s definition, as the shortest ideal description, available to human cognition, of perceived objects or scenes, could prove to be an important notion of cognitive modelling.

  Figure 1: Distribution of chosen structures according to complexity.

Table 1 :

 1 Simple lottery structures of the sample.

	Combinations	Complexity
	1 2 3 4 5 6	3
	34 35 36 37 38 39	6
	10 11 12 44 45 46	11
	7 8 9 37 38 39	12
	8 9 26 27 28 29	12
	10 20 30 31 32 33	12
	1 2 5 6 15 49	14
	. . .	. . .
	14 24 36 38 42 44	26

This is true only for large objects. A better approximation is given by replacing K(x) by K M (x), which is the shortest input of M that gives x as input.

Definitions for the complexity of numbers may slightly vary, but those differences are irrelevant to our purpose here.

The complexity of 0 is not necessarily 0, especially if it is perceived as following 9, in which case the complexity is rather 3.5.

Note that in general, it is difficult to verify that subjective probabilities, as defined by (6), comply with Kolmogorov axioms, as they concern events that are not incompatible. For instance, the expected event of five non-related digits is given probability 1, but it includes all more particular events such as 33333.

Acknowledgments

Special thanks to Hayette Soussou who helped us in the design of the experiment and in carrying out the tests.