Florence De

La Barrière

Guillaume Druart

Nicolas Guérineau

Frédéric Champagnat

Aurélien Plyer

Gilles Lasfargues

AND Serge Magli

Florence De La Barrière
email: florence.de_la_barriere@onera.fr

Compact multichannel infrared camera integrated in an operational detector dewar cooler assembly

Keywords:

INTRODUCTION

Nowadays, military and civilian applications require miniaturized optical systems. Optical architectures have to be compatible with a very low budget in terms of mass, volume and power consumption. These constraints are so demanding that downscaled versions of traditional optical systems with a single optical axis are reaching their limits because of a loss in the number of resolved points in the image [START_REF] Lohmann | Scaling laws for lens systems[END_REF]. Inspired by the compound eyes of invertebrates [START_REF] Völkel | Miniaturized imaging systems[END_REF], multichannel imaging seems to be a promising approach to address this issue. Several types of multichannel architectures have been proposed. Two different approaches can be distinguished. On the one hand, each channel captures a low-resolution image of the overall field of view (FOV), which is known as the thin observation module by bound optics (TOMBO) principle [START_REF] Tanida | Thin observation module by bound optics (TOMBO): concept and experimental verification[END_REF][START_REF] Shankar | Thin infrared imaging systems through multichannel sampling[END_REF][START_REF] Portnoy | Design and charac-terization of thin multiple aperture infrared cameras[END_REF][START_REF] Venkataraman | PiCam: an ultra-thin high performance monolithic camera array[END_REF][START_REF] Carles | Compact multi-aperture imaging with high angular resolution[END_REF]. If the spectral content of the subimages is non-redundant [START_REF] Papoulis | Generalized sampling expansion[END_REF], super-resolution techniques can be applied to retrieve the high frequency content of the scene [START_REF] Tanida | Thin observation module by bound optics (TOMBO): concept and experimental verification[END_REF][START_REF] Park | Super-resolution image reconstruction: a technical overview[END_REF][START_REF] Kitamura | Reconstruction of a high-resolution image on a compound-eye imagecapturing system[END_REF][START_REF] Nitta | Image reconstruction for thin observation module by bound optics by using the iterative backprojection method[END_REF][START_REF] Choi | Signal-processing approaches for imageresolution restoration for TOMBO imagery[END_REF][START_REF] Kanaev | TOMBO sensor with scene-independent superresolution processing[END_REF][START_REF] Shi | Small-kernel superresolution methods for microscanning imaging systems[END_REF]. On the other hand, each channel images a different part of the scene with a reduced FOV [START_REF] Duparré | Artificial apposition compound eye fabricated by micro-optics technology[END_REF][START_REF] Duparré | Microoptical telescope compound eye[END_REF][START_REF] Duparré | Thin compound-eye camera[END_REF][START_REF] Stollberg | The Gabor superlens as an alternative wafer-level camera approach inspired by superposition compound eyes of nocturnal insects[END_REF][START_REF] Druart | Demonstration of an infrared microcamera inspired by Xenos Peckii vision[END_REF][START_REF] Brückner | Ultra-compact close-up microoptical imaging system[END_REF][START_REF] Brückner | Thin wafer-level camera lenses inspired by insect compound eyes[END_REF][START_REF] Meyer | Optical Cluster Eye fabricated on wafer-level[END_REF][START_REF] Li | Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera[END_REF][START_REF] Laycock | Multi-aperture imaging device for airborne platforms[END_REF][START_REF] Laycock | Miniature imaging devices for airborne platforms[END_REF]. The optical axes of the different channels are tilted with respect to each other. The images are then stitched together with an image processing technique in order to reconstruct a final image. Most of the concepts cited above operate in the visible spectral range, and only few address the infrared spectral range (refs. [START_REF] Shankar | Thin infrared imaging systems through multichannel sampling[END_REF][START_REF] Portnoy | Design and charac-terization of thin multiple aperture infrared cameras[END_REF][START_REF] Carles | Compact multi-aperture imaging with high angular resolution[END_REF] work in the long-wave infrared (LWIR) band).

Our field of interest addresses cooled infrared detectors working in the mid-wave infrared (MWIR) band. Infrared focal plane arrays (IRFPA), based on HgCdTe technology, are very sensitive detectors which need to be cooled at a temperature around 90 K. That is why they are integrated in a Detector Dewar Cooler Assembly (DDCA), which is a vacuum chamber closed by a window. A cold shield and a cold diaphragm limit the angle of view of the IRFPA, in order to avoid background emission from the external roof of the Dewar at ambient temperature. Our work aims at miniaturizing optical systems associated with cooled IRFPAs; to do that, we have to integrate the optics directly inside the DDCA. The Dewar is a sealed and cooled environment traditionally dedicated to the IR detector. Specific constraints are linked to the demanding cooled environment: the cryogenic optical architecture has to be as simple as possible and it must have a minimal size. Moreover, the op-tical and mechanical mass added in the cold shield has to be limited, so that the cool-down time could be comparable to the one of a traditional DDCA without optical parts, and the system could remain compatible with operational constraints.

Our approach first consists in integrating a very simple system in the cold shield: it relies on a traditional lens which images the scene with a good optical quality [START_REF] Druart | Towards infrared DDCA with an imaging function[END_REF]. The miniaturization process can be extremely emphasized by integrating the optical system directly on the IRFPA. To achieve extreme compactness, several design rules can be applied: a detailed description of these design strategies can be found in a previous paper [START_REF] De La Barrière | Design strategies to simplify and miniaturize imaging systems[END_REF]. Miniaturization can be achieved by designing multichannel optical systems. We made a cryogenic wafer-level camera whose total track length is equal to 4.08 mm only [START_REF] De La Barrière | Compact infrared cryogenic wafer-level camera: design and experimental validation[END_REF]. However, it is associated with a laboratory IRFPA, and integrated in a laboratory Dewar. Our goal is to adapt the previous wafer-level design to the constraints of a SCORPIO detector integrated in an operational DDCA commercialized by SOFRADIR [29]. In a previous paper [START_REF] De La Barrière | Development of an infrared ultra-compact multichannel camera integrated in a SOFRADIR's Detector Dewar Cooler Assembly[END_REF], we briefly described a new optical and mechanical design which is adapted to a commercial DDCA. In section 2 of this paper, we give more details about the optical design of this prototype. In section 3, we show the optomechanical design. In section 4, we measure the distortion by scanning a point source in front of the camera. The distortion measurements reveal that the shifts between the sub-images differ from the theoretical expected ones, due to technological constraints. We present a new image processing method, called SR Viewer, described in section 5, which is more adapted to the camera than the traditional "shift and add" method. We also simulate errors in the optical architecture to show how irregular sampling can be predicted. We measure the MTF with the bar targets method, with frequencies higher than the Nyquist frequency, which illustrates the quality of the SR Viewer image processing. By controlling the whole image formation chain, from optical design to image processing, this paper addresses the interests and the limitations of multichannel systems.

OPTICAL DESIGN OF THE COMPACT INFRARED CAMERA A. Starting point with a single lens

We have to limit the additionnal mass to be cooled in order not to increase the cool-down time. Thus, as a starting point, we want to design a simple infrared camera made of a single lens integrated in the Dewar. We want to use a SCORPIO off-theshelf detector [29], which has the following features : 640 × 512 pixels, a sampling pitch equal to 15µm and a fill factor equal to 1. The field of view will be equal to 120 • on the diagonal of a 512 × 512 pixels detector (i.e. 10.9mm). To assure correct radiometric performances, we choose a f -number equal to 3. The paraxial focal length is equal to 3.1mm. In practise, to limit astigmatism, the pupil has to be moved away from the lens instead of being on it. After optimization with the Zemax optical software, we obtain the landscape design illustrated in Figure 1, with an entrance pupil in front of the lens. Its main characteristics are given in Table 1. The surface in front of the detector is conic and diffractive (in order to correct lateral chromatic aberration). The lens is made of Germanium. Its thickness on the optical axis is equal to 1.5mm.

For manufacturing purposes, and to be able to use thin optics, we want to keep the lens as simple as possible. That is why only one complex surface (conic and diffractive) is allowed, the other is kept spheric. The modulation transfer function (MTF)

Pinhole diameter 2mm

Optical mass of the lens 0.346g of the optical system is illustrated in Figure 2, until the Nyquist frequency of a focal plane array with a sampling pitch equal to 15µm, i.e. 33.3cy/mm (the MTF of the pixel is not taken into account). The optical system is not limited by the diffraction: it is not so easy to correct optical aberrations while keeping the system as simple as possible. We want to improve this starting point by looking for a system which would have a better optical quality while still relying on a single optical component.

B. Design of a compact multichannel camera

One of the strategies to achieve extreme compactness consists in decreasing the focal length of the optical system [START_REF] De La Barrière | Design strategies to simplify and miniaturize imaging systems[END_REF]. In order to compensate for the loss of resolved points, replicating an undersampled miniaturized system while providing a subpixel shift between adjacent channels is an efficient solution. We still want to use the SCORPIO detector. In the following, we are going to determine how many channels are necessary to retrieve information with an image processing method, called superresolution. In this paper, what we mean with super-resolution is reducing the sampling pitch, which allows maintaining the number of resolved points with theoretical shifts. For cooled MWIR detectors made in HgCdTe, the diffusion length of the carriers [START_REF] Berthoz | Modeling and characterization of MTF and spectral response at small pixel pitch on Mercury Cadmium Telluride[END_REF] reduces the MTF of the pixel, as illustrated in Figure 3. Figure 3 represents (i) the MTF of a 15µm pitch square pixel, it is a sinus cardinal with its first zero at spatial frequency 66.6cy/mm, (ii) the MTF due to the diffusion length of the carriers, which is a Lorentz function, and (iii) the product of the two latest MTF. The global MTF after the cut-off frequency is reduced by a factor 2.9 (from 0.215 without diffusion to 0.075 with diffusion). Trying to retrieve frequencies higher than the pixel cut-off frequency is not so interesting because the lobes have a small amplitude. As we only want to retrieve frequencies up to twice the Nyquist frequency, only 2 × 2 channels are necessary. Moreover, in order to use the maximum pixels available on the SCORPIO IRFPA, the best solution is to have only the theoretical necessary number of channels (2 × 2), and not to increase this number to avoid losing resolved points in the final processed image.

The subpixel shift between two adjacent images has to be

C. Impact of technological constraints on optical design

As we merge the optics and the detector to make an ultracompact camera, technological constrains linked to manufacturing and integrating the optical module near the detector have to be taken into account while designing the system. In previous work [START_REF] De La Barrière | Compact infrared cryogenic wafer-level camera: design and experimental validation[END_REF], the wafer-level camera is totally integrated on the IRFPA: it is made of two stages of Silicon lenslet arrays. The distances between the two lenslet arrays and between the optical module and the detector are equal to a few hundreds of micrometers. The opto-mechanical assembly relies on an innovative way to integrate the optical module directly on the ceramic of the detector. Indeed, Indium bumps are used as spacers between the two stages of lenslet arrays and between the detector and the optical module.

In the current optical design, the detector format increases by a factor of four compared with the one used in the previous demonstrator. Therefore, the focal length of the optics doubles and the back focal length increases too. The distances between some optical components and the detector are greater than 700µm (this technological limit is mentioned in previous work [START_REF] De La Barrière | Compact infrared cryogenic wafer-level camera: design and experimental validation[END_REF]) even if we place the cold filter at equal distance between the detector and the array of lenses. Using Indium bumps as spacers can no longer be applied and the opto-mechanical design needs to be adapted: we use the cold shield as an optical mount, as explained in Section 3.

As the optical architecture is integrated in the cold shield near the detector, it has to be adapted to thermo-mechanical constraints. A study of the SCORPIO detector shows that Germanium has better properties to fulfil the latest requirement than Silicon. Therefore, we choose Germanium as the optical material of the lens. We carry out an optimization process on a single channel using Zemax. The optical layout of a single channel is illustrated in Figure 4. It is simply made of a pinhole and a Germanium plano-convex lens, whose curved face has an aspheric profile. The camera is ultra-compact, since its total track length is lower than 3.8mm. Table 2 sums up the optical characteristics of the multichannel system. Figure 5 illustrates the optical MTF (the pixel MTF is not taken into account). By comparing with Figure 2, we can notice that the optical quality is better than the system with a single lens, as the design is nearly limited by the diffraction. The optical design could be improved in the following ways to reduce the optical mass in the cold shield. Firstly, a thinner plano-convex lens (less than 1 mm) could be obtained by using high-order Fresnel lenses, as shown in previous work [START_REF] Grulois | Reduction of material mass of optical component in cryogenic camera by using high-order Fresnel lens on a thin germanium substrate[END_REF]. Secondly, the cold filter may be deposited directly on the plane surface of the lens (the difficulty may be that the rays for the maximal field of view angle have a high angle of incidence on the plane surface).

If we used the Scorpio detector with a single channel system (as described in section 2A with the starting point), we would obtain a single image on 640 x 512 pixels, sampled with a pitch equal to 15µm (pixel pitch of the detector). With our multichannel design, after image processing such as superresolution, and if the system was perfect, we would obtain a single image with 512 x 512 pixels with an equivalent sampling pitch of 7.5µm = 15µm/2. It would be possible to obtain such resolution with a single channel optical system only if we had infrared detectors with a pitch of 7.5µm, which is not the state-of-the-art today at Sofradir.

FABRICATION OF THE INFRARED ULTRA-THIN CAM-ERA

The Germanium lenslet array was fabricated using diamondturning techniques, which enable the manufacturing of the aspheric profile and of the multichannel design. The manufacturing has been carried out by the French company Savimex [33]. Then, an anti-reflective coating has been deposited on the lenslet array.

We have adapted the design of the cold shield in order to integrate the optical elements (pinhole array, lenslet array and cold filter) near the detector. Small walls have been provided at the top of the cold shield to limit rays with an incident angle greater than the field of view (120 • ). The opto-mechanical design is illustrated in Figure 6. The overall height of the cold shield is lower than 6 mm, whereas the height of traditional cold shields is between 20 mm and 30 mm. The mass of the optical component (lenslet array with its optical and mechanical parts) is lower than 0.7g. Thus, the lenslet array is heavier than the single lens of Fig. 1, but the additional optical mass is nearly compensated by the reduced mechanical mass, as the cold shield is less high. The cool-down time of the camera is comparable to the one of an equivalent DDCA without an imagery function: it is equal to 6 minutes with a THALES Cryogenics RM3. Thus, the camera remains fully compatible with operational constraints. Designing a simple optical system (based on a pinhole array and a lenslet array only) and integrating it in an adapted cold shield leads to a very low additional optical and mechanical mass to be cooled. Then, the whole system has been integrated in an operational DDCA commercialized by SOFRADIR: it is illustrated in Figure 7.

CHARACTERIZATION OF THE ULTRA-THIN CRYO-GENIC INFRARED CAMERA

We performed MTF and NETD measurements. Details on the measurements method and on results can be found in previous work [START_REF] De La Barrière | Development of an infrared ultra-compact multichannel camera integrated in a SOFRADIR's Detector Dewar Cooler Assembly[END_REF]. The main conclusions are that the four channels have similar MTF curves, which means that they are identical from an optical point of view. The quality of the optical system is almost limited by the diffraction, as expected from the theoretical study. The NETD of the whole camera (detector and optics) is equal to 17mK for 50% well-fill and a scene at 293K, which is close to the NETD of the detector alone. The background current due to the instrument is very low; indeed, we measure a very low background current, around 3pA only, because the whole optical system is cooled. In this section, we go deeper in the characterization of the miniaturized infrared camera. We measured the relative illumination in the field and the distortion.

A. Relative illumination in the field

We illuminate the Dewar window with a flat blackbody at 293K: we obtain the image illustrated in Figure 8. We can clearly distinguish the four channels of the camera. We plot the normalized illumination as a function of height in the image plane. We compare this graph to the theoretical data extracted from the Zemax software, as shown in Figure 9. For each subimage, the illumination decreases from the center to the edge, as expected. This evolution is the same for the four channels. We can notice that the decrease is more important on the measured data than on the theoretical one: the decrease on the experimental data is equal to 30% at 1.5mm from the center of the channel, whereas it is only equal to 20% for the same point on the theoretical data. This difference may be explained by Fresnel losses linked to a very high incident angle (60 • ) on the first surface of the small lenses. Zemax does not take into account Fresnel losses while calculating the relative illumination.

B. Distortion measurement

In order to measure the distortion of the cryogenic infrared camera, a resolved point source is scanned with a regular step within the field of view of the camera. It can be localized precisely on the detector by calculating its barycentre for each position. This measurement is performed simultaneously on the four channels of the camera. The relative position of the point sources in the FOV is calculated with respect to the optical centre of each channel. The four measured distortion cartographies are superimposed in Figure 10 and compared to the theoretical data provided by the Zemax software. The distortion is measured on a FOV equal to 69.8 • × 84.54 • . The amount of distortion D is calculated as follows:

D = √ ( x r -x p ) 2 + ( y r -y p ) 2 √ x p 2 + y p 2 , ( 1
)
where ( x p , y p ) are the coordinates of an image point of a system without distortion (black triangles in Figure 10(a)) and (x r , y r ) are the coordinates of an image point produced by the camera (color symbols in Figure 10(a)). In order to measure the maximum amount of distortion, we consider the coordinates of the extreme points at the corners. The maximal distortion is equal to 29.9% in the top right hand corner of the cartography, 30% in the bottom right hand corner, 31.1% in the bottom left hand corner, and 31.3% in the top left hand corner, whereas the maximal distortion calculated by Zemax is equal to 27.4% for this field of view. Therefore, the measured amount of distortion fits correctly the theoretical data.

C. Discussion on tolerances and regular sampling of the scene

In Figure 10, the distortion cartographies seem to be globally identical for the four optical channels. However, zooms on several areas for different FOV values highlight that the distortion slightly differs within the FOV, as illustrated in Figure 11. These differences may have the following explanations. The tolerance on the pitch between the channels, and the tilt between the lenslet array and the detector imply that the sampling on the optical axis is not the regular one expected from the theory. Moreover, optical aberrations are not identical for the four channels, since each channel has been machined independently by diamond turning: the sampling of the scene differs over the FOV (cf. Fig. 11(b)). Therefore, we have to use a more complex algorithm (called SR Viewer and presented in section 5) than a basic shift-and-add technique which may be employed as a first intention. This prototype highlights that the tolerance on the tilt between the lenslet array and the detector has to be excessively tight in order to obtain the regular theoretical sampling of the scene. In Figure 12, we plot how the sampling varies with the tilt between the lenslet array and the detector.

As soon as a small tilt angle is introduced, the sampling differs from the regular one. We can even see that for a tilt angle of 0.11 • , two of the four channels become redundant, which means that the diversity between the channels is not sufficient to enable the reconstruction of a higher resolution image. For our prototype, we measured a tilt angle equal to 0.46 • . The theoretical sampling corresponding to a tilt angle of 0.46 • is illustrated in Figure 12(d). Moreover, there is also a tolerance on the pitch between the channels. This means that, in most cases, the sampling obtained with multichannel systems will be irregular.

To sum up, in practise, it is almost impossible to obtain a regular sampling of the scene because of the tolerances of optical components. To our knowledge, this has never been mentioned in literature on multichannel optical systems. This irregular sampling over the FOV makes the basic "shift and add" algorithm inefficient, and a more complex processing has to be employed.

IMAGE RECONSTRUCTION A. Elements on image reconstruction algorithm SR Viewer

We have developed a super-resolution algorithm, called SR Viewer, which combines the differences between the subimages to retrieve spatial frequencies higher than the Nyquist frequency of a detector with a sampling pitch equal to 15µm (33.3cy/mm).

SR Viewer has been used to process all the subimages of a single frame in order to yield a super-resolved image. It is based on minimization of the following criterion involving a term which matches the experimental data plus a regularization term:

J(I HR ) = ∑ m ϕ d (I LR m -DW m BI HR ) + λ ( ϕ r (∇ x I HR ) + ϕ r (∇ y I HR ) ) , (2)
where I LR m denotes one of the 4 observed subimages, I HR denotes the sought SR image whose dimensions are those of I LR m multiplied by a SR factor L. The observed image model involves a warp matrix W m that models the parallax motion between I LR m and I LR 1 . The latter sets the reference geometry: in other terms W 1 is equal to the identity matrix. In practice, W m is based on optical flow estimates provided by the fast algorithm eFOLKI [START_REF] Plyer | Massively parallel Lucas Kanade optical flow for real-time video processing applications[END_REF]. Matrix B is a 2D convolution operator that accounts for sensor blur, modeled as a Gaussian point spread function. Matrix D is a subsampling operator by the factor L on both dimensions. ∇ x I HR (resp. ∇ y I HR ) denotes first-order, non centered finite difference gradients along horizontal (resp. vertical) direction. Penalizing functions ϕ d is the Huber function [START_REF] Rey | Introduction to Robust and Quasi-Robust Statistical Methods[END_REF], while ϕ r is the Fair function [START_REF] Rey | Introduction to Robust and Quasi-Robust Statistical Methods[END_REF]. These functions are parametrized by a scale parameter s: for values larger than s, ϕ r and ϕ d have a linear behavior, while it is quadratic for values lower than s. The use of such non quadratic functions enables robustness to motion estimation errors for the data term and edge preserving for the regularization term. The trade-off between data term and regularization is tuned by the regularization parameter λ. Criterion (2) relies on an approximate image model where motion is applied after blur, such a model is strictly valid only for translation motion, but applies in practice in wider contexts [START_REF] Létienne | Fast Super-Resolution on Moving Objects in Video Sequences[END_REF] [START_REF] Hardie | A Fast Image Super-Resolution Algorithm Using an Adaptive Wiener Filter[END_REF]. Optimization of convex criterion (2) is performed using a fixed-step gradient descent, in practice ten iterations are enough. All the steps involved are highly parallel and the overall algorithm SRviewer has been implemented on GPU architecture. The examples of Figures 13 and14 have been processed in 130 ms on a Nvidia Quadro M2000M.

B. Retrieval of spatial frequencies higher than the Nyquist frequency

We imaged a scene in a laboratory setting; the distance between the camera and the scene is equal to 2.5m. The scene is composed of a target (illuminated by a flat blackbody at the temperature of 305K) with a diagonal orientation. We make a twopoint non-uniformity correction (NUC) on each raw image acquired by the camera. This NUC correction ensures to avoid non-uniformity response between the pixels, otherwise, it is an issue when we intend to combine images. Figure 13(a) shows one raw subimage. As the subimage is undersampled, we can see a checker-board pattern instead of the target diagonal bars. Figure 13(b) is an interpolation of subimage (a). As expected, a simple interpolation does not allow to retrieve information, since the checker-board pattern is still visible. Figure 13(c) is the image obtained after super-resolution with the information contained in the four optical channels of the camera. We retrieve interesting spatial frequencies since the bar targets appear. However, some defaults on the transition between black and white zones are still visible, this is due to the irregular sampling. Figure 13(d) is the image after super-resolution obtained with the information contained in the four optical channels of the camera and temporal acquisition as well. We add diversity of information by acquiring a temporal sequence of images while moving slightly the camera in front of the scene. With this additional temporal diversity, some of the defaults visible in Figure 13(c) are corrected. When the sampling is irregular, it is difficult to have a good image quality after super-resolution with only 2 × 2 channels. However, designing a 3 × 3 multichannel system to improve the robustness of the super-resolution algorithm results in a loss of resolved points. The solution is therefore to design a 2 × 2 multichannel system and to add diversity by acquiring a temporal image sequence. The temporal acquisition of images is four times faster than what could be done with a single channel camera.

In order to access higher spatial frequencies, we imaged the same scene with increased distances D between the camera and the scene. The spatial frequency of the bar target is given by Equation 3:

ν = N cy L × D, ( 3
)
where ν is the spatial frequency in cycles/radian, N cy is the number of cycles of the bar target, in our case, N cy = 7, L is the dimension of the target (in m), D is the distance between the target and the camera. Therefore, when D increases, it gives access to higher spatial frequencies. Images after processing are shown in Figure 14. Spatial frequencies higher than the Nyquist frequency can be retrieved with a satisfying contrast (for images (a), (b) and (c)). According to the measured MTF [START_REF] De La Barrière | Development of an infrared ultra-compact multichannel camera integrated in a SOFRADIR's Detector Dewar Cooler Assembly[END_REF], the contrast goes to zero at 66cy/mm (pixel cut-off frequency), then the contrast of the bar target is too low for images (d) and (e). We measured the bar target contrast on the images after image processing, and we compared it to the theoretical MTF given by Zemax (taking into account the filtering of the pixel) and to the experimental MTF measured with the spot scan method. This is illustrated in Figure 15 for spatial frequencies up to the cutoff frequency of the 15µm pixel pitch. We can see that the contrast measured on the bar target images clearly fits the MTF of the system. It clearly highlights contrasts higher than 0 for spatial frequencies greater than the Nyquist frequency. The optical quality of the multichannel system is much better than the one of its single channel equivalent (shown in Fig. 2). The contrast goes to zero at the cut-off frequency of the pixel.

CONCLUSION

We have designed, manufactured and characterized an ultracompact cryogenic multichannel infrared camera, working in the [3.7µm -4.8µm] spectral range. It is associated with a standard SCORPIO detector and integrated in a DDCA commercialized by SOFRADIR. The total track length of the optical module is lower than 6mm only and the optical elements are directly integrated in the cold shield. The cool-down time is comparable to the one of an equivalent DDCA without an imagery function, because in the way we miniaturized the optical system, the additional optical and mechanical mass to be cooled is very low. The camera has a wide FOV of 120 • , and is thus well suited for surveillance applications. In the MWIR band, due to the diffusion length of carriers in HgCdTe, it is not worth trying to retrieve spatial frequencies higher that the cut-off frequency of the pixel (twice the Nyquist frequency). 2 × 2 channels are only necessary; adding more channels would result in a loss of resolved points in the final image, that is why we prefer keeping the highest number of resolved points by limiting the number of optical channels. Tolerances on the optical system lead to irregular shifts between the subimages. That is why the algorithm has to be more complex than a basic shift-and-add technique. Some redundancy exists between the four channels, and it becomes necessary to make temporal acquisitions of images to improve image quality after super-resolution. The optical quality of the channels is satisfying and fits the theoretical data. Cooling the optical system provides a very low NETD compared to traditional infrared cameras whose optical parts are outside the Dewar. These developments show that integrating optical functions near the IRFPA can be fully compatible with industrial and operational constraints. Within future work, associating an IRFPA with an array of lenses could be used to build more complex architectures in order to address various applications such as multispectral imaging or 3D vision.

This work was sponsored by the Defense Procurement Agency (DGA) of the French Ministry of Defense.

Fig. 1 .

 1 Fig. 1. Optical layout of the starting point using a single lens. The different colors stand for different field angles.

Fig. 2 .

 2 Fig. 2. MTF of the optical system illustrated in Fig. 1, extracted from Zemax, for different field points (the different colors stand for different values of the FOV). Black curve: diffraction limit, blue: MTF on axis, green: MTF for 7.6mm FOV tangential, green dashed: MTF for 10.9mm FOV sagittal, red: MTF for 10.9mm FOV tangential, red dashed: MTF for 5.43mm FOV sagittal.

Fig. 3 .

 3 Fig. 3. Simulation of the pixel MTF for a pixel pitch equal to 15µm and a fill factor equal to 1, the MTF due to the diffusion length of carriers in HgCdTe, and the global MTF. The global MTF contains poor information after the cut-off frequency of the pixel (66cy/mm here).

Fig. 4 .

 4 Fig. 4. Optical layout of one channel of the ultra-thin infrared camera. The different colors stand for different field angles. Length units are mm.

Fig. 5 .

 5 Fig. 5. MTF of the optical system illustrated in Fig. 4, extracted from Zemax, for different field points (the different colors stand for different values of the FOV). Black curve: diffraction limit, blue: MTF on axis, green: MTF for 45 • FOV tangential, green dashed: MTF for 45 • FOV sagittal, red: MTF for 120 • FOV tangential, red dashed: MTF for 120 • FOV sagittal.

Fig. 6 .

 6 Fig. 6. (a) Opto-mechanical design of the cryogenic ultra-thin camera, (b) Section of the cold shield.

Fig. 7 .

 7 Fig. 7. Photograph of the ultra-thin infrared camera integrated in an operational DDCA commercialized by SOFRADIR.

Fig. 8 .

 8 Fig. 8. Image acquired while illuminating the camera with a flat blackbody at 293K.

Fig. 9 .

 9 Fig. 9. Relative illumination as a function of height in the image plane, (a) measured data on the four channels (each colour corresponds to a channel), (b) theoretical data extracted from the Zemax software.

Fig. 10 .

 10 Fig. 10. (a) Superposition of the distortion cartographies measured in the four optical channels (in color) and a regular grid (black triangles), (b) Theoretical distortion data provided by the Zemax software.

Fig. 11 .

 11 Fig. 11. (a) Reproduction of Fig. 10(a) with squares of different colors selecting different FOV, (b) Zooms on differences between the four optical channels for different FOV.

Fig. 12 .

 12 Fig. 12. Variation of sampling in the reconstructed image with the tilt angle, (a) theoretical case: tilt angle=0 • and regular sampling, (b) tilt angle=0.05 • and irregular sampling, (c) tilt angle=0.11 • and 2 redundant subimages, (d) tilt angle=0.46 • (case of our prototype).

Fig. 13 .

 13 Fig. 13. (a) Raw subimage, (b) Interpolated subimage, (c) Image after super-resolution obtained with the information contained in the four optical channels of the camera, (d) Image after super-resolution obtained with the information contained in the four optical channels of the camera and temporal acquisition as well.

Fig. 14 .

 14 Fig. 14. Image of a scene in a laboratory setting for various distances D between the camera and the scene, corresponding to various spatial frequencies: (a) D = 4m (40cy/mm), (b) D = 4.5m (44cy/mm), (c) D = 5m (49cy/mm), (d) D = 5.5m (54cy/mm) and (e) D = 6m (59cy/mm).

Fig. 15 .

 15 Fig. 15. MTF measured on the bar targets images: it fits well the theoretical and experimental MTF curves.

Table 1 . Optical characteristics of the simple infrared camera based on a single lens illustrated in Figure 1.

 1

	Property	Value
	Size of optics module (L x W x T)	10.9mm × 10.9mm × 9.7mm
	Field of view	120 •
	f -number	3
	Focal length	5.9mm
	Number of pixels	512 × 512 (diagonal)
	Clear diameter of lens	7.4mm

Table 2 . Optical characteristics of the infrared multichannel camera based on the channel illustrated in Figure 4.

 2

	Property	Value
	Size of optics module (L x W x	lower than 8mm × 8mm ×
	T)	6mm
	Field of view	120 •
	f -number of one channel	3
	Focal length of one channel	1.96mm
	Number of channels	2 × 2
	Thickness of small lenses on the optical axis	1mm
	Number of pixels per channel	256 × 256
	lenslet array pitch	3.8mm
	Pinhole diameter	0.6mm
	Optical quality	Limited by the diffraction
	Shift between adjacent channels 7.5µm
	Expected number of resolved points after image processing	512 × 512
	Mass of the optical component (lenslet array)	lower than 0.7g


DAVID HUME AND RATIONALITY IN DECISION-MAKING: A CASE STUDY ON THE ECONOMIC READING OF A PHILOSOPHER

Keywords: Hume, economic philosophy, rationality, decision, passion, emotion, desire, preference, will, choice. JEL classification: B11, B31, B41, D01

This paper shows that Hume's theory of passion, such as elaborated mainly in book II of the Treatise of Human Nature (1739-40) and in the Dissertation on the Passions (1757), gives rise to a conception of the decision process which challenges the canonical approach to the rationality of decision, as rationality of preferences or rationality of choice. It shows that when adopting a Humean perspective, rationality is not embodied as consistency requirements of individual behaviour, but may emerge as a possible outcome of some dispositions of our mind, which make the world inhabited by our emotions.

Introduction

This paper is an unintentional product of an interrogation on the reasons why, as economists, we burden ourselves with the reading of past philosophers, whereas we already have a hard job in dealing with acknowledged economists of the past. The question might seem a bit rude and, for most of us, answering it would be superfluous: as economists, we do not need to answer it to go on working on philosophers. Rude though, this question is raised hereafter concerning the works of David Hume, as a special case study. It gave rise to four typical answers, which have in common to investigate alternative ways of borrowing something from a philosopherhere, from Hume. Something that we can use as economists:

1. A transposition of a way of reasoning, which had historically made sense for economistsand sometimes, still makes sense.

2. A direct contribution to economic ideas of his time.

3. An influence on acknowledged economists.

A challenge to contemporary economics.

 Phare, University Paris 1 Panthéon-Sorbonne -106-112, boulevard de l'Hôpital -75647 Paris Cedex 13 -France. E-mail: andre.lapidus@univ-paris1.fr.

Obviously, these four issues refer to possible approaches of the works of philosophers like Hume. Recalling their content should help emphasizing the last of them, which is retained in the following of the paper.

1. A transposition of a way of reasoning. The first issue echoes Karl Pribram's understanding of the history of economic reasoning, as applications of principles of reasoning elaborated elsewhere, namely in philosophy [START_REF] Pribram | A History of Economic Reasoning[END_REF]. This led him to stress the use of a nominalist method of reasoning, or of associationism, in Hume's criticism of Baconian economics (using Pribram's words) and in his contribution to the rise of utilitarianism. In the same way, Hume's careful attention to practical matters concerning economic life, his interest in the figure of the merchant, as emphasized, for instance, by M. [START_REF] Schabas | Hume on Money, Commerce, and the Science of Economics[END_REF], can be viewed as an effect of an empiricist approach to economic matters.

2. A direct contribution to economic ideas of their time. The second issue echoes J.A. Schumpeter's position, when he claimed that " [Hume's] economics has nothing whatever to do with either his psychology or his philosophy" (Schumpeter 1954: 474n.). This means that we would not be misleading when arguing that although a philosopher, we have good reasons to move away from Hume's Treatise of Human Nature (1739-40) and consider Hume to be an economist, when he deals, in some of the short texts introduced in his Political Discourses since 1752 onwards to be finally published as parts of the Essays Political, Moral, and Literary (1777), with such questions as interest, public expenditure, money, and the balance of payment. It is in this way, for instance, that J.A. [START_REF] Frenkel | The Monetary Approach to the Balance of Payments -Essential Concepts and Historical Origins[END_REF] drew on Hume's price-specie flow mechanism in order to introduce historically the monetary approach to the balance of payments, or that M. [START_REF] Friedman | The Quantity Theory of Money[END_REF] opened his review article on the quantity theory of money.

3. An influence on economists. Typical of this issue is the attention paid to, say, the respective influence of Hegel and Kant on Marx and Walras. Such influence might be either complex, like in the case of Marx, when considering the transformations of the Hegelian concept of alienation into the Marxian concept of exploitation, or limited to a mere transposition, like for Walras who, drawing on a popularized knowledge of Kant, uses the distinction between phenomenon and noumenon in order to view exchange as a natural fact which constitutes the object of pure economics. Obviously, this issue is also involved in the discussion of Hume's influence on Adam Smith when, for instance, looking for the roots of the latter's moral philosophy or philosophy of knowledge, we are induced to go back from Smith's Theory of Moral Sentiments (1759) to books II or III of Hume's Treatise (1739-40), to his second Enquiry (1751) and to the Dissertation (1757), or (with fewer textual evidence) from Smith's History of Astronomy to Hume's book I of the Treatise and to his first Enquiry (1748). Notably, such perspective focuses on a philosopher's work only insofar as it sheds special light on the work of the possibly influenced economist, which constitutes the determining stake.

4.

A challenge to contemporary economics. The question is no more this of knowing how a philosopheragain, David Humedid contribute to the shaping of the economic knowledge of his time, but how he might contribute to the shaping of the economic knowledge of our time. This means that when Hume writes as a philosopher, explicitly on those questions which Schumpeter, in the above quoted passage, regarded as irrelevant from an economic point of view, he challenges what we know in economics, analytically and methodologically. This claim was at least indirectly supported by such authors like Ken Binmore through the specific part he granted to David Hume in the history of ideas. Since the 1980's, he tried to explain that various passages from the Treatise on Human Nature, book III, like the one about the "[t]wo men, who pull the oars of a boat, [and] do it by an agreement or convention, though they have never given promises to one another" (Hume 1739-40: 3.2.2.10 1 ; see also 3.2.5.9.

The same ideas are repeated, more than ten years later, in the second Enquiry; see Hume 1751: App. 3.8.) provide a solution to some well-known difficulties in game theory concerning such topics like the selection of an equilibrium, or the implementation of a sustainable cooperation as equilibrium in repeated games (see Binmore 1994: 27-34;2005: 4).

Whatever our doubts about the question of knowing whether Hume did, or did not, have a firm enough intuition of a line of answers to these difficulties, I'd like to focus on the fact that Binmore's argument rests on the idea that Hume considered differently the relation between the self and the other, and that he aimed at investigating the consequences of such difference upon the interaction between individuals. From a methodological viewpoint, this paper might be considered in line with Binmore's approach to Hume. But analytically, it focuses not on interactions between individuals, but on decision-making in relation to rationality. Relying chiefly on books I and II of Hume's Treatise, it will be shown that he induces us to view rationality of both preferences and choices not as embedded in the axiomatic basis of decision-making, but as a possible outcome of a decision process. This will lead to proceed in the following way. The first step consists in recalling the characteristics of what can be regarded as a canonical understanding of rationality in decisionmakingi.e., rationality of preferences and rationality of choice (section 2). This allows, in a second time, to assess the type of modifications of this canonical view, which comes along with a Humean approach of decision-making, as modifications regarding the premises and properties of preferences and choice, giving birth to rationality as a possible outcome (section 3). Keeping this objective in mind explains that in the following, emphasis is laid on the consequences of a specific reading of Hume, rather than on the way such consequences are obtained (see, on this issue Diaye and Lapidus 2005a). The conclusion (section 4) allows showing the precise scope of Hume's challenge to the canonical theory of decisiona 1 Reference to Hume's works published by Oxford University Press are given according to the divisions of the edition: for the Treatise, book, part, section and paragraph numbers; for the Abstract, paragraph numbers; and for the two Enquiries, section and paragraph numbers. challenge which regards the way we address the issue as economists, and our philosophical implicit assumptions.

A canonical approach: rationality assumed

The canonical way of considering rationality in the simplest framework of decision (that is, leaving aside time, risk or uncertainty) follows a representation of the tastes of a decisionmaker through a preference relation and a choice function which respectively satisfy some meaningful properties. The preference oriented side is now quite common, and rests on comparisons within pairs of alternatives, whereas the choice function side is a bit less familiar, and amounts to the selection of one or some alternatives among subsets of the set of choices. Preference and choice might be viewed as either simultaneous or mutually exclusive primitives in the canonical approach. But in each case, the relations between the conditions of rationality of preferences and of choice (comparatively, the second might be viewed as the looser) constitutes a crucial issue.

Rational preferences

The preference side of the canonical representation was introduced in economic literature through successive attempts to identify the axiomatic basis which allows a binary relation of preference to be represented by a utility functionwhat G. Debreu achieved in 1954. Such was the case for R. [START_REF] Frisch | Sur un Problème d'Economie Pure[END_REF][START_REF] Frisch | New Methods of Measuring Utility[END_REF], F. [START_REF] Alt | Über die Messbarkeit des Nutzens[END_REF], and H. [START_REF] Wold | A Synthesis of Pure Demand Analysis, Part II[END_REF]. Though N. Georgescu-Roegen (1954: 119) traced back to Frisch's first published paper what he called the 'modern' (i.e. post-Paretian) approach to choice, as distinct from preferences, he seems to have been more interested in the expression of preferences and of their relative intensity in order to build a function of utility. Frisch gave them the presumably first axiomatisation (see [START_REF] Bjerkholt | Ragnar Frisch's Axiomatic Approach in Econometrics[END_REF][START_REF] Bjerkholt | Ragnar Frisch's Axiomatic Approach in Econometrics[END_REF]Dupont-Kieffer 2013: 27-35), consisting in completeness (Frisch's "axiome de choix"), transitivity ("axiome de coordination"), and in an axiom on the effect of infinitesimal variations ("axiome d'addition"). This allowed him to accomplish what he called "le rêve de Jevons" (the dream of Jevons) (Frisch 1926: 79), that is, presenting a formal statement of the representation of preferences and of their intensities by a utility (cardinal) function. Yet, the influence of Frisch's paper, written in French in 1923 during a stay in Paris and published in a Norwegian journal, was rather poor. Although more available to the scientific community at the end of the 1950's thanks to its republishing in Metroeconomica (1957), it was not translated into English before 1971, together with the translation from German of the paper of his continuator Franz [START_REF] Alt | Über die Messbarkeit des Nutzens[END_REF] (Kelly 1987: 44). So that, as Arrow put it, "whenever I saw a 𝑈 for a utility function I translated to a preference ordering" (Kelly 1987: 46). The consequence was a most impressive Rand Corporation's working paper [START_REF] Arrow | The Possibility of a Universal Social Welfare Function[END_REF]), presented at the December 1948 meeting of the Econometric Society in Cleveland: this paper introduced not only a first exposition of the impossibility theorem, but also the main ideas of the canonical approach to decision making (Arrow 1948: 3-9)a binary relation of preference with the properties of a complete preorder, and a choice function which satisfies consistency requirements, like already in the revealed preference approach pioneered by P.A. Samuelson in 1938. It therefore announced Arrow's following works: his article (Arrow 1950: 331-3) and his book (Arrow 1951: 11-7) on social choice and, some years later, his article on individual choice and preferences [START_REF] Arrow | Rational Choice Functions and Orderings[END_REF]. Rationality of preferences fits Arrow's early statement of the preference relation as a complete preorder 3 . It first depends on a double premise, this of the definition of the set of kürzere Artikel desselben Verfassers) (Alt 1936: 167). On Alt's contribution, see T. Ellingsen 1994 (pp. 121-2) and I. Moscati 2013 (pp. 925-7). 3 A "weak ordering", in the terminology of Arrow 1948: 4;1959: 122. objects to which preferences are applied (the set of reference of choice), and after that, this of the relation of preference itself, expressed by a binary relation over the set of reference of choice:

𝑋 (set of reference of choice) [1] 𝑅 ⊆ 𝑋 2 (relation of preference) [2]
According to [2], preference of 𝑥 on 𝑦 is noted (𝑥, 𝑦) ∈ 𝑅. The alternative and more commonly used notation is 𝑥𝑅𝑦, which reads "𝑥 is preferred to 𝑦" (which does not exclude the possibility that 𝑦𝑅𝑥, in which case 𝑥 and 𝑦 are said "indifferent"). In the canonical approach, the double premise [1] and [2] characterizes an individual, from a decision theory point of view. It concerns this part of the external world, pre-existing to him or to her, potentially subjected to his or her tastes, and these tastes, embodied in a given structure of preference 4 At this point, nothing has been said about the properties of the preference relation, except that it exists, at least for some, if not all, 𝑥 and 𝑦 belonging to 𝑋. Our preferences are said to be "rational" when 𝑅 is a complete preorder, that is, when they satisfy [3] and [4]. Both completeness and transitivity provide possible and not exclusive interpretations for rationality. On the one hand, completeness [3] refers to some kind of cognitive ability, in the sense where the comparability between any elements of 𝑋 is independent of its dimension. This seems rather demanding, since it dismisses any answer of the kind: "Well, I really don't know. If I were indifferent, it would mean that I could toss a coin to decide. But here, it's not that I don't mind having 𝑥 or 𝑦. I really don't know how to compare 𝑥 and 𝑦. I just can't answer…" 5 . Lack of completeness thus appears less as non-4 Though intuitive, the idea that 𝑋 in [1] and in [2], as support of the preference relation, is typically constituted by goods or services, like in ordinary consumer theory, is not that obvious: underlying the notion of value function in D. [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF] for instance, and more generally in reference-dependent models, preferences are not defined on a state, figured by goods and services, but on a difference to a state of references, that is, on gains and losses. There is, of course, no objection to the representation of gains and losses by 𝑋. 5 In a paper rather critical on the rationality assumptions concerning preferences, H. [START_REF] Putnam | Über die Razionalitet von Praeferenzen[END_REF] illustrated the case of incomparability with the example of an individual, called Theresa, who is torn between an "ascetic" (𝑦) and a "sensual" way of life but, in case she chooses the sensual way of life, who prefers as a lover 𝑥 to 𝑧. rationality than as a limitation to rationality, as what H. Simon called "bounded rationality"which amounts to regard "mind as the scarce resource" (Simon 1978: 9). On the other hand, transitivity [4] refers to a consistency requirement, at least when strict preferences are at issue. When such is not the case, we can easily adhere to the idea that, because of an insufficient ability to discriminate between close elements of 𝑋, non-transitivity might occur when indifference is involved. Such restriction amounts to substitute quasi-transitivity for transitivity in [4], so that 𝑅 becomes what R.D. [START_REF] Luce | Semiorders and a Theory of Utility Discrimination[END_REF] called a "semi-order". At least quasi-transitivity, therefore expresses this necessity of the mind, which can hardly be ruled out since it contains the core of the idea of the rationality of preferences.

Rationality comes along

Rational choice

Understanding the rationality of preferences nonetheless allows bypassing an external feature of the characterization of the individual, this of the subset of the set of reference of choice, illustrated by the budget in the theory of consumer behaviour, over which some elements can be chosen. Now, focusing on choices over alternative non-empty subsets of the set of reference of choice, named "contexts of choice" (or "opportunity set", or "budget", or "menus", according to alternative terminologies), rather than on preferences on a set of reference of choice, continues an intuition drawn from Samuelson's path-breaking article on revealed preferences [START_REF] Samuelson | A Note on the Pure Theory of Consumer's Behaviour[END_REF], systematized by Arrow as soon as 1948 (Arrow 1948: 4-5;1951: 15-7;1959: 122-3). In this approach, the primitive of decision moves from the binary relation of preferences over 𝑋 (what Sen 1987 called "binariness") to the function of choice. Here again, the analysis rests on given premisesthe identification of the set of reference of choice, the set of alternative contexts (the "domain of choice") which are made available to the individual, and this of the choice function which is defined over it:

𝑋 (set of reference of choice) [1] 𝐹 ⊆ 𝒫(𝑋)\∅ (domain of choice) [5] 𝐶: 𝐹 → 𝒫(𝑋), 𝑆 ⟼ 𝐶(𝑆) (choice function) [6]
Like in the preference approach, the set of reference of choice 𝑋 is given by [1]. Denoting 𝒫(𝑋) the power set of 𝑋, [5] makes explicit the possible restrictions on the various contexts of choice 𝑆 on which a choice can be made. When all non-empty parts of X are possible contextsthat is, when 𝐹 = 𝒫(𝑋)\∅the domain of choice 𝐹 is said to be "abstract" or "non-restricted". Otherwise, when 𝐹 ⊂ 𝒫(𝑋)\∅, the domain 𝐹 is "restricted". Note, for instance, that assuming that choices are submitted to an income constraint, like in elementary Putnam showed that here, this lack of completeness would also violate negative transitivity (∀𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝑥, 𝑦) ∉ 𝑅 and (𝑦, 𝑧) ∉ 𝑅 ⇒ (𝑥, 𝑧) ∉ 𝑅), otherwise Theresa could not prefer 𝑥 to 𝑧.

consumer theory, means that the domain of choice is restricted to the set of well-known triangles representing the expenses possibilities of the consumer. According to [6], {𝑥} = 𝐶(𝑆) reads "𝑥 is chosen over 𝑆". Note that this is a quite general formulation since, instead of the singleton {𝑥}, we may have, for instance, {𝑥, 𝑦} or even, if 𝐶 is non-selective, ∅. In the same way as in a preference-oriented approach, the choice-oriented approach acknowledges as its starting point something given in order to characterize an individual: the way external objects are made available to him or to her, and his or her possible choice among these alternative sets of objects. And here again, the question of the rationality of choices is raised when adding supplementary properties to the function of choice so simply defined as in [6].

On first view, these properties can be introduced without explicit reference to preexistent preferences. Such was the case since [START_REF] Arrow | The Possibility of a Universal Social Welfare Function[END_REF], with H. [START_REF] Uzawa | Note on Preference and Axioms of Choice[END_REF], and with [START_REF] Arrow | Rational Choice Functions and Orderings[END_REF], who defined a rational choice function by choice-consistency conditions.

Reference to choice-consistency conditions deserves special attention. In his contribution from 1959, Arrow set out five alternative "definitions" of a rational choice function, labelled from C1 to C5, and explored their logical relations, showing the equivalence between C2 and C3, between C1, C4 and C5, and the implication of the first group of equivalent definitions by the second. This puts an end to an elaboration which started in 1948, when he gave one single definition of a rational choice function, which amounted to C4 (later known as Arrow's axiom), and continued with Uzawa (1959) who gave two definitions which amounted to C1 and C2. All these definitions regarded choice (and not preferences) as a primitive, so that whereas the way a rational choice was defined might have been different, it always amounted to internal consistency. Yet, the vocabulary in use concerning the definition of rational choice functions might be misleading. Again in [START_REF] Arrow | Rational Choice Functions and Orderings[END_REF], definition C5 corresponds to the so-called Weak Axiom of Revealed Preferences (WARP) (Arrow 1959: 123), whose initial statement comes from [START_REF] Samuelson | A Note on the Pure Theory of Consumer's Behaviour[END_REF]. But in spite of its name, it didn't involve any pre-existent preferences (see Richter's reservation to Samuelson's terminology using the word "preference" (Richter 1971: 32, n. 4)). Its formulation in Arrow's article means that when I choose x on a context S where y was available and not chosen, there is no other context T in which x and y are available, and where y would be chosen. This amounts to the following statement, which clearly contains no reference to preferences: ∀𝑆 ∈ 𝐹, ∀𝑥, 𝑦 ∈ 𝑆, 𝑥 ∈ 𝐶(𝑆) and 𝑦 ∉ 𝐶(𝑆) ⇒ ∄𝑇 ∈ 𝐹: 𝑥, 𝑦 ∈ 𝑇 and 𝑦 ∈ 𝐶(𝑇) (WARP)

Yet, since M. [START_REF] Richter | Revealed Preference Theory[END_REF][START_REF] Richter | Rational Choice[END_REF], it has become difficult to make a so complete abstraction of preferences in the definition of a rational choice function, though the binary relation is now a property associated to the choice function, and not a representation of the tastes of an individual. We are indebted to Richter for a definition of the rationality of the choice function as the existence of a specific compatibility between our choices and a hypothetical underlying relation of preferencethe rationalizability of a choice function by such hypothetical relation of preference (Richter 1971: 31). This does not imply any comeback from choice to preferences as a primitive; but it requires the introduction of the possibility of a preference relation as a criterion for rationality. A mere possibility: it does not mean that we do have preferences, but that this relation stands for those that we could have. And if such was the case, our choices on each context 𝑆, when rational, would have been in accordance with our preferences on this context. Following Richter, rationalizability hence amounts to: ∃𝑅 ⊆ 𝑋 2 : ∀𝑆 ∈ 𝐹, 𝐶(𝑆) = {𝑥 ∈ 𝑆: ∀𝑦 ∈ 𝑆, (𝑥, 𝑦) ∈ 𝑅} (rationalizability)

[7]

According This explains that a significant part of the literature has been devoted to the search for conditions under which both choice and underlying preferences are rational. For instance, when the domain of choice is unrestricted (resp., restricted), the Weak Axiom of Revealed Preferences (resp., the Houthakker Axiom of Revealed Preferences) is a necessary and sufficient condition to the rationality of choice, the underlying preference relation being also rational. But generally speaking, rationality of choice is less demanding than rationality of preferences.

The canonical approach produced formal statements about both rationality of preferences and rationality of choice. Each of them is now as widely known as it is widely threatened, from about the middle of the twentieth century onward. Calling this approach "canonical" doesn't mean that it should be regarded as the most accepted to decision-making (see Zouboulakis 2014): it is "canonical" only because it conveys a reference against which alternative approaches can be, and use to be, assessed. In a sense, it plays the same role as, say, perfect competition, when the issue is the understanding of imperfect competition. In the following, it will therefore constitute the appropriate starting point to account for the importance of the challenge to which we are invited by the works of David Hume.

6 In his 1966 paper, Richter still held that 𝑅 should be itself rational, that is, that preferences are complete and transitive. However, he noted in a footnote the possibility of a more general conception of rationality, where 𝑅 would be non-complete and non-transitive (Richter 1966: 636, n. 4).

A Humean perspective: Rationality as a possible outcome

Let us turn back to Hume. The decision process inferred from Hume's theory of passions, developed mainly in book II of the Treatise and in the Dissertation, has been formally described in Diaye and Lapidus (2005a). Whereas this paper does not conclude with the incompatibility between Hume's conception of a decision process and the rationality of

Which premises for preferences and choices?

What David Hume called "passions" or "impressions of reflection" can be conveniently approached by what we would name today "emotions", but restricted to the modalities ofthat is, to the ways to livethe sensations of pleasure and of pain. At the very beginning of book II of the Treatise, they are presented "as arising either from the original impressions [of pleasure and pain; A.L.] or from their ideas" (Hume 1739-40: 2.1.1.3). Round-trips between Hume's teaching on the passions and canonical decision theory allow showing the possible consequences of a Humean approach 1) on the nature of the primitive for decision-making, 2) on the stability of the preference structure, and 3) on the respective places of preferences and choice as well as this of the sets over which they are defined.

The nature of the primitive for decision-making

An often quoted passage from the Treatise emphasizes the asymmetry between reason and passions, presenting the first as the "slave" of the second (Hume 1739-40: 2.3.3.4). A result of this asymmetry is the picture of emotion-driven individuals (see [START_REF] Lapidus | The Possibility of a Welfare Policy in a World of Emotion-Driven Individuals: A Humean Point of View[END_REF], drawn all along Hume's philosophical works. The relevant emotions here belong to what he called the "direct" (by opposition to "indirect") passions, which "arise immediately from good or evil, from pain or pleasure" (Hume 1739-40: 2.1.1.4) and converge toward "desire" (or its opposite, "aversion") and "will"the latter coming immediately before action (see part 3 of book 2 of Hume 1739-40, and section 1 of Hume 1757) 7 . A comparison between this picture and the current one, drawn from canonical decision theory, rests on a transposition from Hume's desire and aversion to preferences, and from will to choice. Diaye and Lapidus (2005a: 94-6) give textual evidence which corroborates such transposition.

7 The direct passions presented in the Treatise and in the Dissertation are joy and grief, hope and fear, desire and aversion, and volition or will. On Hume's theory of passions, P.S. Árdal's book from 1966 still remains an impassable introduction.

An immediate conclusion is that whereas canonical decision theory does not really settle the question of the primitive for decision-makingeither preferences or choice -, or even avoids asking it in such terms, the coexistence, within Hume's theory of passions, of both desire and will (though, as will be seen hereafter, Hume seems to hesitate counting the will among the passions), shows that both preferences (desire) and choice (will) can be regarded as the primitives for decision-making.

Nonetheless, it is obvious that this way of replacing desire and will in a decisiontheory framework overlooks a typical feature of Hume's explanation of action, the role of "belief", as a transitional device which gives to an idea a part of the "force and vivacity" (Hume 1739-40: 1.3.7.5) of the original impression of pleasure or pain, thus allowing desire or aversion. Yet, the relation between belief and desire and their respective parts in the determination of action constitute a classical philosophical issue. From a Humean viewpoint, this issue typically opposes those who, like M. [START_REF] Smith | The Humean Theory of Motivation[END_REF] or E. [START_REF] Radcliffe | Hume on the Generation of Motives: Why Beliefs Alone Never Motivate[END_REF], claimed each in his or her own way that belief alone couldn't give rise to action because it has a mind to world direction-of-fit, and those who, like B. [START_REF] Stroud | Hume[END_REF], considered belief as somehow similar to a passion, thus liable to be at the origin of action; or like A. [START_REF] Baier | A Progress of Sentiments -Reflections on Hume's Treatise[END_REF] for whom reason has some kind of access to the determination of action through belief; or like R. Cohon (2008: 30-62), who stressed the role of belief in expected pleasures, which cause motivating passions. I do not intend to underestimate the importance of this question (see Lapidus 2010, Diaye and Lapidus 2012), but whatever the answer, my purpose here is to focus on some of its consequences which might be taken up disregarding the explicit link to belief and pleasure.

And indeed, it remains that desire is related to the will and to decision. It is only this limited aspect of Hume's contribution which constitutes the subject matter of the discussion favoured in this paper.

A changing structure of preferences

In contrast with the implicit assumptions of canonical decision theory, but in accordance with the intuitive view of any one who is not a professional decision-theorist or economist, a second difference between the canonical view on rational decision and Hume's conception is that for the latter, the individual is changing in his or her emotional state. Hume finds in the so-called "rule of double relation"relation of ideas, and relation of impressions (Hume 1739-40: 2.1.4;Hume 1757: 144-5)the principle which explains such changes from one emotional state (one passion) to another, from one configuration of desire and choice to another. The "relation of ideas" leads from the object of the first passion to the object of the second. It follows the "natural relations" (resemblance, contiguity, and causality; Hume 1739-40: 1.1.4.1-4) which Hume regarded as the elementary components which allow the working of our mind 8 . For instance, when discussing the case of pride associated to the organization of a feast, Hume draws on contiguity and causality in order to explain that the "feast"the object of the initial direct passion of joy -is related to the "self"the object of pride, the indirect passion which follows joy -for the one who is "the master of the feast" (Hume 1739-40: 2.1.6.2). The second element of the double relation is the "relation of impressions", based on the natural relation of resemblance. In the example of the feast, the pleasure which is at the origin of the direct passion of joy is associated to the one that arouses the indirect passion of pride: the pleasure which I feel in participating in a feast meets the one of organizing it. The working of the double relation is explicitly referred to when Hume tries to explain the transformation of our desire and will: "[A] suit of fine cloaths produces pleasure from their beauty; and this pleasure produces the direct passions, or the impressions of volition and desire. Again, when these cloaths are consider'd as belonging to ourself, the double relation conveys to us the sentiment of pride, which is an indirect passion; and the pleasure, which attends that passion, returns back to the direct affections, and gives new force to our desire or volition, joy or hope" (Hume 1739-40: 2.3.9.4).

An individual, moved by this engine constituted by the double relation, can no more be represented both through given and unchanging preferences, and through a given and unchanging choice function, defined over a given and unchanging domain of choice. This does not lead to conclude that preferences [2] and choice [6] have become meaningless, but rather that they are fleeting, following the changes in our emotional statewhich each of us already knows, especially if he or she has forgotten to be an economist or a decision-theorist.

The respective places of preferences and choice, and of the sets over which they are defined

The third difference between Hume's perspective and the canonical approach is less intuitive.

In the latter, the set of reference of choice is given first; and then only a binary preference relation is defined, or a domain of choice and a choice function. Now, with Hume, this order (the set of reference first, then the primitives, preferences or choice) becomes irrelevant. In keeping with Hume, objects cannot be considered, in our minds, pre-existent data: rather, they are constructed by our desires and, more generally, by our emotions. So that the objects possibly submitted to our preferences and choice are not objects that would exist independently of our impressions, as pointed to us by our reason, but the only objects the existence of which we acknowledge through our emotions: 8 In the enthusiasm of the Abstract, Hume concludes that the natural relations "are really, to us, the cement of the universe, and all the operations of the mind must, in a great measure, depend on them" (Hume 1740: 35).

"Where the objects themselves do not affect us, their connexion can never give them any influence; and 'tis plain, that as reason is nothing but the discovery of this connexion, it cannot be by its means that the objects are able to affect us" (Hume 1739-40: 2.3.3.3).

Other objects could exist for other individuals or for an external observer, but for the agent who does not desire them, they simply do not exist. This is a way to say that what is given first is not something as general as a set of reference of choice, but a contextual preference relation 𝑅 𝑆 . And if so, the context of choice 𝑆 can be easily identified as its support 9 : 𝑆 = {𝑥, 𝑦: (𝑥, 𝑦) ∈ 𝑅 𝑆 or (𝑦, 𝑥) ∈ 𝑅 𝑆 } (generation of the context of choice)

[8]

And as our desires change along with our emotional state, our contextual preferences also change, therefore changing the context of choice:

𝑅 𝑆1 , 𝑅 𝑆2 , … , 𝑅 𝑆𝑖 → 𝑆 1 , 𝑆 2 , … , 𝑆 𝑖 (contextual preferences change) [9]
As a result, when the context of choice changes according to [9], the general set of choice also changes, by union to the successive contexts:

𝑋 𝑖 = 𝑋 𝑖-1 ∪ 𝑆 𝑖 (set of reference of choice) [10]
The result of this first investigation might seem a bit disappointing. The simple and elegant presentation of the canonical approach seems to have vanished: instead of a given set of reference of choice on which preferences and choice are built, we are facing the immediate emergence of contextual preferences, which come along with a corresponding context of choice and, therefore, with a revised set of reference of choice. But all that is constantly submitted to transformations due to emotional change, so that the provisional conclusion would be: we have preferences, but they are always moving; they are not submitted to any control, so that their greatest elements on the moving contexts of choice might be anything, and can be explained by no regularity depending on the properties of the preference relation.

Rationality as an outcome

Nonetheless, the picture of such an unstable individual, left as it is, would be most unfaithful to Hume's analysis. What we can understand from Hume's decision process shows three types of regulating devices, whose effect is to stabilize it into a resulting configuration of preferences and choice. These devices consist 1) in a mental device which makes our 9 Rigorously speaking, regarding 𝑅 𝑆 as a binary relation would require the precondition of a universal set of objects on which it is defined (see Diaye and Lapidus 2005a: 97), so that it is by abuse of notation that we keep on using 𝑅 𝑆 to denote a binary relation defined on 𝑆.

contextual preferences complete, 2) in a special link between preferences and choice, and 3) in a stabilizing property.

Completeness of contextual preferences

The above discussion of the "connexion" between objects (Hume 1739-40, 2.3.3.3) concluded an analysis developed in the Treatise of the possibility for reason, through the natural relation of causality, to make that some objects exist in our mind on its sole basiswhich Hume denied. But the way the argument was developed before this conclusion is worth being quoted more extensively. Hume was talking about the aversion attached to an object:

"'Tis also obvious, that this emotion rests not here, but making us cast our view on every side, comprehends whatever objects are connected with its original one by the relation of cause and effect.

[…] Tis from the prospect of pain or pleasure that the aversion or propensity arises towards any object: And these emotions extend themselves to the causes and effects of that object, as they are pointed out to us by reason and experience" (Hume 1739-40: 2.3.3.3).

This amounts to saying that the contextual preferences 𝑅 𝑆 are connected (see Diaye and Lapidus 2005a: 96-7), in the sense that each element of 𝑆 is linked to any other element through a chain of theses preferences, built by the natural relations of the mind.

Interestingly, this property of connection doesn't come alone. It takes place within an analysis of natural relations, introduced in book 1 of the Treatise, which provides also a rule of extension of the natural relation in the imagination and, therefore, of our desire or preferences:

"That we may understand the full extent of these relations, we must consider, that two objects are connected together in the imagination, not only when the one is immediately resembling, contiguous to, or the cause of the other, but also when there is interposed betwixt them a third object, which bears to both of them any of these relations" (Hume 1739-40: 1.1.4.3).

This is a way of saying that where three elements 𝑥, 𝑦 and 𝑧 of 𝑆 are such that 𝑥 and 𝑦, on the one hand, 𝑦 and 𝑧, on the other hand, are related by contextual preferences 𝑅 𝑆 , 𝑥 and 𝑧 also are linked by 𝑅 𝑆 . Now (Diaye and Lapidus 2005a: 97), it can be shown that when connection and extension both prevail as properties of the contextual preferences 𝑅 𝑆 , this means that they are also complete:

∀𝑥, 𝑦 ∈ 𝑆, (𝑥, 𝑦) ∈ 𝑅 𝑆 or (𝑦, 𝑥) ∈ 𝑅 𝑆 (contextual completeness) [11] Considering completeness, even when it is contextual, as the result of the natural working of the mind contrasts not only with canonical decision theory, but also with Simon's view, as recalled above: since our cognitive ability is supported by our emotions and their movements, claiming with H. [START_REF] Simon | Rationality as Process and as Product of Thought[END_REF] that the occurrence of completeness is prevented by the limits of this cognitive ability, would become meaningless.

Contextual compatibility between preferences and choice

The question of knowing whether the will is to be considered among direct passions is far from completely solved. On first view, some important textual evidence favours the idea that the will should be included among direct passions: this is explicitly the case at the end of the third part of Book II of the Treatise on the will and the direct passions, in the section dedicated to the direct passions (Hume 1739-40: 2.3.9.2), as well as in that of the Dissertation (Hume 1757: 139). On the contrary, other passages suggest that the will does not belong to the direct passions. It was included neither in the list which appears at the beginning of the first part of book II of the Treatise (Hume 1739-40: 2.1.1.4), nor at the beginning of its third part (Hume 1739-40: 2.3.1.1). Still more, in this last section, Hume writes that it is "not comprehended among the passions" (Hume 1739-40: 2.3.1.2).

Such seemingly contradictory statements require some explanation. The issue, for Hume, was highly polemical since it was this of free will, in which he supported a conception of freedom as "liberty of spontaneity" (that is, lack of violence in the determination of action), by contrast to the "liberty of indifference" (which implies that actions do not obey to necessity, that is, are independent from desires), which he rejected (Hume 1739-40: 2.3.2.1-2) 10 . Suppose the will is excluded from the list of passions. It might therefore be understood as a faculty liable to determine action and to take precedence over the passions, allowing free will in the sense of the liberty of indifference. This clearly argues in favour of the opposite position, that is in the inclusion of the will within the list of passions. However, such inclusion is not fully satisficing, since it leaves open the possibility that the will is another determinant of action, with its own informational content. Now, Hume had explained at length how the determining role of pleasure and pain is distorted by belief in order to give rise to desire and aversion 11 , and there is no more room for a supplementary and hardly understandable distortion of desire which would give birth to different will. According to Hume, the will says nothing else than the desire already saidexcept that it is to be followed by action when it is not altered by a transformation of passions:

10 "Few are capable of distinguishing betwixt the liberty of spontaneity, as it is called in the schools, and the liberty of indifference; betwixt that which is opposed to violence, and that which means a negation of necessity and causes. The first is even the most common sense of the word; and as it is only that species of liberty which it concerns us to preserve, our thoughts have been principally turned towards it, and have almost universally confounded it with the other" (Hume 1739-40: 2.3.2.1). See the discussion of Hume's position on free will by N. Kemp Smith (1941: 439-41), P. Árdal (1966: 87-9), and P. Russell (2008: 225-38), who also replaces Hume's argument within the religious debates among his contemporaries. 11 On the place of Hume's beliefs in decision-making, see supra, p. 11.

"DESIRE arises from good consider'd simply, and AVERSION is deriv'd from evil. The WILL exerts itself, when either the good or the absence of the evil may be attain'd by any action of the mind or body" (Hume 1739-40: 2.3.9.7).

In other words, the will only converts the prescriptions of desire into action. This is a sufficient ground to explain Hume's uneasiness facing the place of the will: there are good reasons to include it in the passions, but as such, it would add nothing to themso that the basis of such inclusion is not that firmly established.

But leaving apart the question of knowing if the will is or is not a genuine passion, its characterization leads to the conclusion that from a Humean point of view, if we choose according to what we will in a context of choice, what has been chosen fits what we contextually prefer the most, according to our desire: 𝐶(𝑆) = {𝑥 ∈ 𝑆: ∀𝑦 ∈ 𝑆, (𝑥, 𝑦) ∈ 𝑅 𝑆 } (contextual compatibility)

[12]

Such compatibility might be viewed as an attenuated kind of rational choice like in [7], but restricted to one context of choice and to contextual preferences.

Till the whole circle be completed: allowing free choice

Though contextual completeness [11] and compatibility [12] show an embryonic content in terms of rationality of preferences and of choice, the assumption of changing preferences seems far more important. This change means a revision of our general dispositions, which concerns more than isolated contextual preferences and the corresponding context of choice, as stated above in [9]. The story and the meticulous description of such changes is a typical exercise from Hume. In the second Enquiry, for instance, when he was discussing something apparently quite different (the "qualities useful to ourselves") he came to examine the comparative effects of close and distant agreeable objects. And he shows that when such object becomes nearer, this does not only mean that our desire for it increases, but also, in Hume's words, that our "general resolutions" have changed (Hume 1751: 6.15). The phrase is successful. It means that a change in specific desires (contextual preferences) 𝑅 𝑆 i gives birth to a change about which we have not that much information, except that it occurs and that it is general, thus involving general preferences 𝑅 𝑋 i to such an extent that they can supersede the contextual preferences 𝑅 𝑆 i in case of conflict concerning the choices on 𝑆 𝑖 . This also leads to modifications of the domain of choice 𝐹 𝑖 , as described in Diaye and Lapidus (2005a: 101-2),

as consequences of the effect on choices of the revised preferences. But generally speaking, whatever the precise characteristics of these modifications, they yield:

(𝑅 𝑋 𝑖-1 , 𝐹 𝑖-1 ) Preference stabilization [14], jointly with contextual compatibility between preference and choice [12], show that there is a moment when the decision process is achieved, which means that decision is not prevented from occurring by emotional changes, and that it now respects stabilized preferences and choice. In other terms, it fulfills desire and will. Such fulfillment is obtained when the decision process is not interrupted by any external circumstance, and is left to its own evolution toward a stable configuration. This means that the described decision process leads to a free choice, in the precise sense of the already mentioned "liberty of spontaneity", which Hume had endorsed in the Treatise, by contrast with the "liberty of indifference". The first Enquiry is still more explicit than the Treatise 12 :

"By liberty, then, we can only mean a power of acting or not acting, according to the determinations of the will; that is, if we chuse to remain at rest, we may;

if we chuse to move, we also may. Now this hypothetical liberty is universally allowed to belong to every one, who is not a prisoner and in chains" (Hume 1748: 8.23).

One aspect of the contrast between this liberty of spontaneity and the liberty of indifference, ruled out by Hume, should be stressed, when dealing with free choice resulting from Hume's 12 As pointed out by P. Russell (2008: 375, n. 65), Hume's vocabulary has changed between the Treatise and the Enquiry. The "liberty [...] universally allowed" from the latter corresponds to the "liberty of spontaneity" in the former. From a strategic point of view, this change helped transforming an opposition between the supporters of different kinds of liberty, into an opposition between the supporters of a conjunction of liberty and necessity and those who deny it.

decision process: the choice is free not according to an external observer like, for instance, from a libertarian viewpoint which would mean liberty of indifference, but only according to the feeling of the decision-maker himself or herself. With a different formalization, it has been shown in Diaye and Lapidus 2005a that this decision process pictured by relations from

[9] to [14] gives birth, when completed, to the two types of rationality in decision-making identified above, in section 2:

1. Rationality of choice: when 𝑖 is greater than or equal to 𝑛, the choice function 𝐶(𝑆) defined on 𝐹 𝑖 is rationalizable by the general preference relation 𝑅 𝑋 𝑖 .

2. Rationality of general preferences: when 𝑖 is greater than or equal to 𝑛, 𝑅 𝑥 𝑖 is a complete preorder, that is both complete and transitive. This amounts to the claim that, in the specific meaning of the phrase in a Humean framework, a decision process leading to free choice makes this choice rational, along with the decisionmaker general preferences.

Conclusion: From Philosophy to Economics

On first view, the rationality of choice and preferences seems to be a rather strong statement: in spite of a different treatment, the transposition of Hume's teaching in the words of decision theory brings about the same result as canonical theory does. But on second view, it becomes evident that such conclusion would be misleading. In canonical theory, rationality is assumed, as consistency hypotheses which come at the very beginning of the description of the properties of preference and choice: such is the case with rationality of preferences, insofar as completeness and transitivity are concerned; such is also the case with rationality of choice, which might be approached through various consistency assumptions, like the weak axiom of revealed preferences, or Houthakker's axiom of revealed preferences, or the so-called properties 𝛼 or 𝛽. In all cases, rationality is embodied as consistency requirements of individual behaviour. On the opposite, Hume's initial statements do not recover such initial consistency requirements. It only expresses some disposition of our mind: a way to make the world inhabited by our emotions. See, for instance, what might have appeared as the first expression of minimal rationality content: contextual completeness [11]. It is not given as such, but it is a consequence of the working of our mind, when it appropriates the world through desires, going from one object to another, and relating two objects where they are linked by a third one. The same can be argued about the general conclusions on the rationality of preferences and choices. Such rationality was not presupposed at an initial stage. The only thing that we have to regard as granted is that our emotions are changing, among which our preferences, inducing us to revise them again and again; and since our mental world at last quiets down, after a certain point they do not need any more to be revised. If this point is reached, it means that we are in the conditions of a free choice. Then, rationality stands out, but now as a consequence of the working of our mind in the decision process, no more as its prerequisite.

As economists, this challenges directly the way we take up rationality in decisionmaking. It leads us to give up questioning its relevance as normative behavioural assumptions, and, conversely, we are prompted to question the effects of all the external or internal circumstances which might deviate or interrupt the functioning of the mental process which opens onto decision: instead of being alternative views of individual behaviour, rationality and irrationality become the outward signs of the absence or of the presence of a hindrance to the course of the decision process.

Yet, Hume's challenge is also more indirect, in that it concerns implicit but determining philosophical positions which govern the way we are presenting a problem as economists, like in the opposition made by Sugden in his seminal paper of 1991, between already Hume and Kant. Here, it compels us to question what usually remains to us out of reach, if not out of interest: our understanding of an individual whose description boils down to the specification of the part of the external world which constitutes his or her set of choice, and of his or her preferences. Such spontaneous understanding expresses a philosophical position. Hume obviously offers another one, in which the part of the external world on which choices are to be performed is constructed by our preferences. We may agree or disagree with this philosophical position. But after all, this is not the most important: what really matters is that as a by-product of an investigation on Hume, we acknowledge that we have a philosophical position, that the latter is responsible of what we do as economists and that, for this very reason, it has to be questioned.

  with the two following properties of the relation of preference, which make it a complete preorder and constitute the well-known axiomatic basis of the standard theory of consumer behaviour: ∀𝑥, 𝑦 ∈ 𝑋, (𝑥, 𝑦) ∈ 𝑅 or (𝑦, 𝑥) ∈ 𝑅 (completeness) [3] ∀𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝑥, 𝑦) ∈ 𝑅 and (𝑦, 𝑧) ∈ 𝑅 ⇒ (𝑥, 𝑧) ∈ 𝑅 (transitivity) [4]

  preferences and choice (for an opposite view, see Sugden 2005 and the answer of Diaye and Lapidus 2005b), it nonetheless challenges the canonical conception, in their premises ([1], [2],[5],[6]) and in the part played by rationality ([3], [4],[7]). This shows that a concern, as economists, about a possible change in our understanding of the individual as a decisionmaker is far from being analytically neutral: such change, carried on by David Hume, comes along with a modification in the representation of decision-making.

  to [7], a choice function 𝐶 defined on 𝐹 is said rational when we can imagine a preference relation 𝑅 with support 𝑋 such that on each context of choice 𝑆, what is chosen over this context is also what would have been preferred, according to 𝑅. Numerous examples illustrate cases where a consistency condition is unsatisfied, so that no preference relation could, like in [7], rationalize the choice (see, for instance, A. Sen 1993: 500-3). But even when the choice is rational according to [7], it is clear that since nothing is said concerning

the properties of the preferences 𝑅 which rationalize the choice, they might be non-complete (not satisfying [3]) or non-transitive (not satisfying

[4]

), therefore non-rational 6 .

  The revision procedures [9], [10] and[13] (which constitute a simplified version of the "UPDATE procedure" in Diaye and Lapidus 2005a) do not contain, by themselves, a regulating device which would take them to an end. It is when discussing the part played by the association of impressions in the mechanism of the double relation, that Hume suggests Till the whole circle be completed: Hume's expression suggests a cumulative process, in which grief, disappointment, anger, envy and maliceall the possible emotions linked by the natural relation of resemblancefollow one another again and again, till they do not change anything to our general resolutions, that is, to our desires and will. This amounts to saying that there exists a step, denoted 𝑛, after which the new contextual preferences which appear as in

	that there is a moment when the revision is completed:
	"Grief and disappointment give rise to anger, anger to envy, envy to malice,
	and malice to grief again, till the whole circle be compleated" (Hume 1739-40:
	2.1.4.3; see also Hume 1757: 145).
	𝑅 𝑆 𝑖 → (𝑅 𝑋 𝑖 , 𝐹 𝑖 ) (general preferences, domain of choice)	[13]

[9], preserve general preferences in [13]:

∃𝑛: ∀𝑖 ≥ 𝑛, (𝑥, 𝑦) ∈ 𝑅 𝑆 𝑖 ⇒ (𝑥, 𝑦) ∈ 𝑅 𝑋 𝑖 (preference stabilization)

[14]