Fig. 1. Examples of stimuli of numerosity 9, 30 and 100 belonging to the 3 stimulus sets. Stimuli were originally yellow on a black background. Stimulus set 1: total occupied area and total luminance are constant, thus dot size decreases with numerosity and array density increases with numerosity. Stimulus set 2: total occupied area and dot size are constant, total luminance and array density increase with numerosity. Stimulus set 3: array density and dot size are constant, total luminance and total occupied area increase with numerosity. In the three sets the mean dot size and mean total luminance are equal. Mean density is about twice larger in the 3rd set. Consequently, total occupied area is about twice smaller in the 3rd set.

Fig. 2. Responses of the five non-calibrated participants. Horizontal axis: stimulus (n); Vertical axis: response (R). The top row shows the distribution of the responses, using a gray color scale to represent the logarithm of the frequency of occurrence of the response R for each stimulus n. On the second and third lines, mean responses are presented, first on a linear plot, and then on a log–log plot, where both horizontal and vertical axes have been log-transformed. If the response function are power shaped as predicted in the theory part, they should appear as a straight line on a log–log plot. On the last row, error bars representing standard deviation of the responses are added to the log–log mean response plot. The property of scalar variability predicts that standard deviation bars should all appear of the same length on this graph.

Fig. 3. Responses of the 3 groups of induced participants (same format as Fig. 2). All groups were presented with an inducer trial which they were told contained 30 dots. In the overestimated inducer group, the inducer contained 25 dots; in the exact inducer group it contained 30 dots; in the underestimated inducer group it contained 39 dots.

Fig. 4. General presentation of the model. Numerosities are first encoded on an internal continuum, the mental number line. Activations on the number line are then translated into verbal number words, by means of a response grid: the number line is divided in segments, each corresponding to a different verbal label. The theory section developed here precises the shape of the response grid. The bottom panel represents the distribution of responses emanating from a gaussian distribution of activations on the number line. Bar areas are proportional to the frequency of the responses, and bar widths respect the shape of the response grid. In these conditions, the curve formed by the bars (highlighted in black) follow the initial gaussian activation curve (in light gray).

Fig. 5. Response grids. (a) In the canonical response grid, the criteria lay at the midpoint between the log(n)s. (b) The spontaneous and calibrated response grids are related to the canonical response grid is an affine transformation: the canonical response grid is either stretched or compressed (parameter a), and shifted (parameter b).

Fig. 6. Predicted response matrices. Horizontal axis: stimulus numerosity (n); Vertical axis: verbal response (R). Shades of gray indicate the probability of the response R for the stimulus n. Plain lines give the mean responses, and dotted lines indicate the correct value of the stimulus numerosity (diagonal R = n). Three cases are presented, corresponding to different values of the parameters of the affine transformation applied to the canonical response grid. Compressed response grid: a < 1; Canonical response grid: a = 1; Stretched response grid: a > 1.

Fig. 7. Reconstruction of the activations on the number line. (a) Distribution of activations on the number line, averaged across all participants, and compared with its gaussian fit. The standard deviation of the best-fitting Gaussian curve (R² = 0.99) corresponds to the estimated internal Weber fraction: w = 0.22. (b) Z score of the distribution in (a), for the interval of the internal number line encompassing 90% of the data.

Fig. 8. Evolution of the internal Weber fraction (w, left panel), and the two parameters of the affine transformation applied to the response grid (stretch on the middle panel and shift on the right panel) between the two experimental sessions in Experiment 2. X-axes represent the value of these parameters for the first session, and Y-axes the values for the second session. Doted lines indicate the equality diagonal (first session = second session). Each dot represents one participant: participants presented with a larger inducer in the second session are marked as filled squares, and participants presented with a smaller inducer in the second session are marked as unfilled triangles. Values indicated on the axes correspond to the means for the first and second sessions. The graphs illustrate how each subject possesses an idiosyncratic value of parameters w, a and b, which is reproducible across the two sessions, and also demonstrate that only the parameters a and b are affected by calibration.

List of abbreviations used in the text