Is number a primary perceptual attribute? *

Abstract

In this article, I briefly review evidence accumulated over the past two decades that number should be considered a primary perceptual attribute. I concentrate primarily on adaptation studies, but also show how these have been reinforced with other psychophysical methods, as well as pupillometry, EEG and imaging studies. There have been at least two major challenges to number being a primary perceptual attribute, and to adaptation itself, which I discuss. I conclude speculating on the functional role of adaptation to numerosity perception, and suggesting areas requiring further research.

How to Cite this Article

Burr, D. (2026). Is number a primary perceptual attribute?. i-Perception, 17(3), 1–21. https://doi.org/10.1177/20416695261423067

A Visual Sense of Number

Nearly two decades have passed since John Ross and I published our short paper titled ‘A visual sense of number’ (Burr & Ross, 2008a), where we suggested that number is directly perceived rather than calculated from other sources: ‘we propose that just as we have a direct visual sense of the reddishness of half a dozen ripe cherries, so we do of their sixishness. In other words there are distinct qualia for numerosity, as there are for colour, brightness, and contrast’. This conclusion was based largely on the evidence that number, like other perceptual attributes such as colour and motion, adapts. How has this idea survived the test of time?

John had always been interested in numerosity, from when he acquired then state-of-the art computers (PDP-8) and point-plotting oscilloscopes, decommissioned from NASA's space station in Carnarvon, Western Australia. They could rapidly display arrays of bright points, which he used to study stereopsis and other visual phenomena (e.g., Ross & Hogben, 1974). John was also fascinated by the simpler question: can we spontaneously estimate how many dots are on the screen? He observed that the impression of dot numerosity was immediate, spontaneous and compelling, suggesting that it was sensed directly, rather than based on a complex calculation. This may not seem an outrageous idea now, given the generality and importance of numerosity for survival and everyday life, but at that time vision research was dominated by concepts of spatial frequency, channels and texture: that the system was designed for more pragmatic tasks like number estimation was not high on the agenda.

Sensory systems follow a set of general principles. One is that all tend to follow Weber's Law (discrimination thresholds proportional to base level), at least over a limited range. John's first study (Ross, 2003) with number demonstrated Weber's law for number, reinforcing earlier studies (Burgess & Barlow, 1983; Jevons, 1871), and consistent with numerosity being a perceptual system. Another universal characteristic of perceptual systems is adaptation, underscored by John Mollon's (1974) famous assertion: ‘At its most naked, and within the privacy of the laboratory, the argument runs “If you can adapt it, it's there”’. As we had just published on the spatial selectivity of visual adaptation to duration (Burr et al., 2007), it was fairly natural to test whether numerosity was also adaptable.

Adapting to dense dot fields drastically changes the apparent numerosity of dot fields of moderate density. The effect is stunning (see video 1), and easily replicable, never failing to impress. That no particular dot actually vanishes shows that it is not the dots themselves that are adapted, but the representation of the numerosity, strong evidence that numerosity is represented in some form of global, summary statistic, rather than as individual dots; and this estimate, like other perceptual representations, is modified adaptively by perceptual history.

Emboldened by the compelling effects, John suggested we try the inverse: adapt to small numbers and test moderate numbers. I was resistant to the idea, expecting it to fail, but eventually capitulated and was surprised by the results. While the effect was less dramatic than adapting in the other direction, careful measurements with trained observers showed a robust increase in numerosity after adapting to low numbers (see Figure 1A). The result has since been replicated many times (Aagten-Murphy & Burr, 2016; Aulet & Lourenco, 2023; Zimmermann, 2018), and confirmed by subtle techniques involving observer confidence (Benedetto et al., 2026), to which I will return later.

Figure 1.

Adaptation to numerosity. (A) The effect of adapting to spatial arrays of various numerosities on the perceived numerosity of a 50-dot array of black and white dots. (B) The effect of adapting to sequences of flashes, average rate of 2/s (low) or 8/s (high), on the perceived number of flashes. The normalized difference in slopes gives the adaptation effect. (C) Cross modal and cross-format adaptation for a variety of adaptors (upper labels) and tests (lower labels).

The experiment was timely, coming out as Nieder (2005; Nieder et al., 2002) was describing number-specific cells in the primate, and Piazza et al. (2004) had demonstrated similar neural mechanisms in humans with fMRI adaptation techniques. This all gave credence to the claim that the mechanisms of numerosity estimation are best considered part of our perceptual system.

The Speckled Hen

That adaptation acts on the general impression of numerosity rather than on the single dots is important for many reasons. One is that it addresses the philosophical ‘problem of the speckled hen’, presented by Gilbert Ryle to A. J. Ayer (Church, 1941) to challenge the nature of sensory representation: ‘consider the sense datum yielded by a single glance of a speckled hen: how many speckles does the datum comprise?’. That humans cannot enumerate the number of speckles was considered a major challenge to prevailing philosophical theories about ‘given, direct experiences’. The adaptation studies clearly show that the individual speckles are not ‘enumerated’, but that the total number of speckles is estimated and represented by a summary statistic, and this estimate is subject to contextual effects, like all other perceptual estimates (see also Butterworth, 2008).

It is interesting that Ryle chose number for his challenge, rather than the colour, height or weight of the hen, all equally difficult to gauge with great precision. But number illustrates the paradox best, as it is clearly a discrete digital quantity, yet our perceptual estimates are very approximate. As we have multiple ways of measuring number, including systematic and errorless serial counting, we can easily verify how rough our estimates are, whereas the other senses require photometers, tape-measures or scales to betray their imprecision.

That number estimation is imprecise and essentially noise-limited is further evidence that it is a sensory system. Ayer did not have the concept of noise-limitation in 1940—introduced to neuroscience a few years later, largely by Horace Barlow (1961)—but correctly anticipated that while the hen does have a definite number of speckles, the sense datum has only an imprecise, approximate guess: essentially, he anticipated the approximate number system.

Challenges to the Number Sense

Our first public presentation of the results was at the European Conference in Visual Perception in Arezzo in 2007, causing strong, but varying reactions. Many were impressed with the vividness of the demonstrations, and the novel approach, considering numerosity as a sensory system: but others had major reserves. The main contention was that the effects were due to adaptation of texture density, not numerosity: ‘Rather than requiring that the visual system jump right to a numeric representation, density aftereffects can be explained by earlier visual processes. It is possible to represent a correlate of density as something like statistical kurtosis in the visual image and this may be evaluated at various spatial scales’ (Durgin, 2008). This seemed to us an unnecessarily contortionist route to gauge the number of things in the world, ‘reminiscent of the legendary Australian stockman who when asked to explain his uncanny ability to judge the number of cattle in a herd replied that he counted the legs and divided by four’ (Burr & Ross, 2008b). However, the challenge prompted some interesting research.

Durgin's argument that it is texture rather than number that adapts rested largely on showing that adapting to an array larger in spatial extent than the test (so its density is low) is less effective than adapting to a dense array matched to the test. This, he claims, is evidence that it is density, not numerosity, that counts. However, as we countered in our reply (Burr & Ross, 2008b), like most forms of adaptation, numerosity adaptation is spatially localized (which is why it works at all, reducing the numerosity of only the corresponding patch). What counts is not the total number of dots anywhere on the screen, but those in correspondence with the test stimulus. More recently, others have used this specificity to measure the size of what we can term perceptive fields of numerosity adaptation (Zimmermann, 2018), showing that the estimates of the sizes of perceptive fields tuned to numerosity are compatible with what would be predicted from receptive fields in the intraparietal sulus (IPS) of monkey (Nieder, 2005, 2016).

A recent study by Adriano and Velde (2025) provides very clear evidence that adaptation acts on numerosity rather than texture. Adapting to high-pass, illusory dot patterns, made from the ‘Ehrenstein’ illusion, causes strong adaptation to standard dot patterns. This is interesting for many reasons, but shows that the adaptation is not due to texture, given that there was no overlap in the spatial frequency spectra of the adaptor and test stimuli.

Does the system respond spontaneously to numerosity or density?—given that the two covary, numerosity being the product of density and area. Marco Cicchini et al. (2016) tested which of the three dimensions—numerosity, density or area—is the visual system most sensitive. The task was to identify the odd-one-out in three dot arrays, one differing from the other two, randomly in number, density or area. Figure 2A shows how accuracy varies over the two-dimensional space spanning log-area and log-density (with log-numerosity falling on a diagonal). The two-dimensional psychometric function (passing through the 75% correct point) was elongated along the direction of constant numerosity. This means that sensitivity was highest (shorter distance) when numerosity varied, and lowest when area and density varied inversely, keeping numerosity constant. Numerosity emerged as the most sensitive dimension to be spontaneously sensed. This result was reinforced by various other experiments, including direct reproduction of the size and density of dot patches (Cicchini et al., 2019): again the highest errors were at constant numerosity and the lowest when it varied most.

Figure 2.

Evidence for the spontaneity of numerosity perception. (A) Average proportion of correct identification of the odd-one-out target in three dot arrays: two identical (24 dots within a disk of 7.2° radius), with the target varying in density and area by the amount indicated on the ordinate and abscissa. The two-dimensional psychometric function describes an ellipse oriented along the constant numerosity axis. Discrimination was best (short axis) when the density and area covary to produce changes in numerosity, and worst when they covary to keep numerosity constant. Reproduced from Cicchini, Anobile and Burr (2016). (B). Pupillary response to presentation of four luminance-matched images, illustrated in the icons. The response is given by the difference in pupil diameter while observing black compared with white dot patterns, averaged over 16 observers. The response was highest for 24 isolated dots, and lowest for 18 connected dots, showing that the pupil gain increases with perceived dot numerosity. Reproduced from Castaldi et al. (2021).

A good deal of evidence shows that numerosity is sensed spontaneously, independently of texture. One clear demonstration is the connectedness illusion, reported shortly after our adaptation studies, independently by Franconeri et al. (2009) and He et al. (2009). When pairs of dots are connected by lines (or by more subtle means) to create dumbbells, the numerosity of the dot field seems much reduced (see video 2). Although adding lines increases the physical density by adding further texture, the apparent numerosity decreases compellingly. The connecting lines activate grouping mechanisms, which cause the connected dots to be encoded as a single item, reducing the perceived numerosity: another clear dissociation of numerosity from texture density. Interestingly, when connected patterns are adapted, the pattern of adaptation follows the perceived, rather than physical numerosity (Fornaciai et al., 2016).

Physiological Evidence for the Number Sense

There is a good deal of physiological evidence supporting the existence of the number sense. Piazza et al. (2004) were the first to use fMRI to demonstrate number-selective neural units in parietal cortex of humans, using the selective adaptation technique of Grill-Spector and Malach (2001)—incidentally furnishing direct evidence for the adaptability of the number sense. Since then, this finding has been amply confirmed in many laboratories, most notably by Harvey and collaborators (Harvey et al., 2013), demonstrating a clear topographical organization for the number selectivity. Combining magnetoencephalography with MRI techniques, Piazza's group (Karami et al., 2025b) went on to show that the neural signature for number appeared very rapidly after stimulus onset, preceding the encoding of non-numeric features (such as density) that could otherwise define it. EEG recordings (Fornaciai et al., 2017; Park et al., 2016) also show that the human brain is extremely sensitive to numerosity (more so than to density), and responds very rapidly to numerosity information, as early as 75 ms at occipital sites.

Multivariate decoding studies have been particularly important in understanding neural mechanisms of numerosity perception. Eger et al. (2009) showed that parietal activation patterns for individual numerosities can be reliably discriminated and generalized, despite massive changes in low-level stimulus properties, such as density, showing that numerosity encoding is not reducible to simple visual features. Studies using representational similarity analysis with tight stimulus control (e.g. Castaldi et al., 2019; Karami et al., 2025a) have shown that numerosity explains unique variance in neural responses beyond other quantities. Karami et al. (2025a) further extended these findings using whole-brain searchlight analyses, helping to clarify other brain regions where numerosity information is encoded.

During her Marie Curie fellowship, Elisa Castaldi showed that even the very basic pupillary light response is modulated by numerosity (Castaldi et al., 2021). The pupils respond to arrays of white dots by constricting, and of black dots by dilating. However, the gain of the pupillary response was not driven only by luminance (matched for all conditions), but was higher for patterns with more dots, clear evidence of the salience of numerosity (Figure 2B). When pairs of dots were connected to reduce the apparent numerosity via the connectedness illusion, the gain of the response reduced further. After adaptation by dense or sparse dot arrays, the gain of the pupil light response varied with the apparent (adapted) numerosity, rather than the physical numerosity (Caponi et al., 2024).

Castaldi also showed that for numerosities within the estimation range, observers make very fast ‘express’ saccades towards the more numerous target, with latencies as low as 190 ms (Castaldi et al., 2020). There were far fewer express saccades for numbers within the subitizing range, or for high densities comprising textures.

The most recent research from our laboratory highlighting the salience of numerosity is the rapid ‘implicit learning’ of numerosity. Binda et al. (2025) recorded a pupillometric signature of the learning of the pairing of numerosities of sequential arrays, as observers simply viewed the sequences of stimuli, without performing any task. Although the pairings were too rapid to be noticed, after a few trials of simple observation, the pupil responded clearly to their repetition frequency.

While outside the primary scope of this review, it is worth mentioning the wealth of evidence for the existence of the number sense from comparative studies of many species. Andreas Nieder (Nieder et al., 2002) was the first to record from cells in monkey pre-frontal cortex, selectively tuned for numerosity. This fascinating result has been confirmed many times, and extended to other areas, including ipsilateral parietal cortex (Nieder, 2005, 2016). These and subsequent studies all took great care to control for extraneous variables that covary with numerosity, including density, greatly reinforcing the evidence for the existence of a number sense independent of density, and extending the concept beyond humans.

Number cognition is not limited to primates. Nieder's group have shown similar selectivity for number in neurones in the endbrain of the crow (Wagener et al., 2018). Indeed, numerosity perception seems to be a universal trait amongst the animal kingdom (Butterworth et al., 2018): it has been reported in newly hatched chicks (Lorenzi et al., 2024), zebrafish (Agrillo & Bisazza, 2018), bees (Dacke & Srinivasan, 2008) and many other animals. There is even evidence that the Venus flytrap has a rudimentary system of counting insect steps to minimize false alarms (Böhm et al., 2016).

Importantly, animal studies provide a method of testing whether number perception is innate (or acquired extremely early), as one may expect from a primary sense. Viswanathan and Nieder (2015) measured numerosity selectivity in prefrontal cortex (PFC) and ventral intraparietal area (VIP) in numerically naïve monkeys before and after numerosity training. Both areas showed selectivity for number, before any numerosity training, showing an innate tuning for number. Similar innate tuning has been demonstrated in the endbrain of the crow (Wagener et al., 2018), newly hatched visually naïve chicks (Kobylkov et al., 2022) and even larval zebrafish (Luu et al., 2024). New-born human infants have also been shown to perceive number, visual and auditory (Gennari et al., 2023; Izard et al., 2009).

Numerosity and Texture

Although a good deal of research shows that numerosity is as a salient perceptual attribute, it is clear that texture is also a perceptual quality, to be studied in its own right. Are the two connected, sharing common mechanisms (Dakin et al., 2011; Morgan et al., 2014)? Intuitively, they would seem to be related, transitioning from one to another as number increases: while a few isolated birds do not usually form a texture, a murmuration of starlings certainly can. Is there a transition point, where numerosity becomes texture, and different systems are involved?

Anobile et al. (2014) measured numerosity discrimination over a wide range of numerosities and densities, and observed that at moderate densities, thresholds indeed followed Weber's law (increasing directly with numerosity), but at high dot-densities, they increased with the square root of numerosity. These distinct psychophysical laws point to two distinct mechanisms for high and low densities: the approximate number system for moderate densities, and texture-density mechanisms for items too closely packed to be individuated. The point of transition varies with eccentricity (Anobile et al., 2015), from 2.3 dots/degree² in central viewing to 0.5 dots/degree² at 15 degrees eccentricity, reminiscent of Bouma's law for crowding. This suggests that under crowded conditions, the approximate number system cannot discern the separate items, and gives way to texture-density mechanisms.

In this light, we should consider three mechanisms for processing number (Anobile et al., 2016b; Burr et al., 2017): subitizing, for four or fewer items; the approximate number system, for numerosities greater than four, but where elements can be individuated as distinct items; and texture density mechanisms, for high densities where the individual items are not resolvable.

Many other studies have supported the distinction between numerosity and texture: the connectedness effect is much reduced at high, non-estimable numerosities (Anobile et al., 2017). There are far fewer express saccades to high-density patterns, than in the estimation range (Castaldi et al., 2020). Perceptive field sizes are smaller in the texture range than in the estimation range (Zimmermann, 2018), consistent with receptive field sizes in primary and secondary visual areas, compared with parietal and frontal. Numerosity discrimination thresholds predict math performance in children (Halberda et al., 2008): however, this occurs only for numerosities within the estimation range, not for higher, texture-like numerosities (Anobile et al., 2016a).

A General Multimodal Sense of Number

In his critique of number adaptation, Durgin (2008) astutely observed that ‘cross-modal studies seem a more promising avenue for distinguishing aftereffects of perceived number from retinotopic aftereffects in the early visual analysis of texture density’. Twelve years ago, Roberto Arrighi suggested measuring adaptation to temporal sequences of flashes. Again, I was neither hopeful nor encouraging: but his experiments exceeded all expectations (Arrighi et al., 2014). As shown in Figure 1B, estimation of the numerosity of sequences was highly influenced by the previous sequence: fast sequences of events caused underestimation and slow sequences overestimation. The normalized difference in slopes of the two curves gives an adaptation index.

Not only was the numerosity of sequential flashes adapted, but the effect was completely cross-modal (Figure 1C). Adapting to auditory sequences affect vision and vice versa. Adapting to tactile sequences affects both auditory and visual sequences, and vice versa (Togoli et al., 2020; Togoli & Arrighi, 2021). Importantly, we demonstrated ‘cross-format adaptation’: adapting to sequences of flashes affected the numerosity of spatial arrays, in both directions. Perhaps, as Durgin anticipated, this is the strongest evidence yet for adaptation to visual number without involving texture.

Another young colleague, Giovanni Anobile, pushed the effects further by measuring adaptation to action. Simply tapping the hand slowly or rapidly (with the hand concealed from view and haptic feedback minimized) affected the apparent numerosity of sequences of flashes (Anobile et al., 2016a), and also of sounds (Togoli & Arrighi, 2021). Again, hand-tapping also affected the apparent numerosity of spatial arrays. All this points to a truly generalized sense of number, transcending space and time, sensory modality, and action.

These findings find strong parallels in animal studies. Nieder (2012) described cells in monkey prefrontal and posterior parietal cortices that are selective to numerosity, with the same selectivity to auditory and visual sequences, and later in the endbrain of the crow (Ditz & Nieder, 2020). Similarly, Sawamura et al. (2002) have reported cells in parietal cortex of monkey selective to the number of internally generated movements, irrespective of movement type, a putative neural substrate for the generalized adaptation to action.

Generalization and Specialization

The studies mentioned above show how the sense of number generalizes over space and time and modality. But there is also evidence for its specificity: all types of adaptation showed strong spatial specificity, with effects restricted to the region that has been adapted, pointing to adaptation-defined ‘perceptive fields’ (Zimmermann, 2018). This is, of course, a common property of adaptation, suggesting perceptual rather than conceptual processes (like internal counting). Interestingly, for adaptation to sequences of flashes, the specificity was in external, spatiotopic coordinates, rather than retinotopic coordinates (Arrighi et al., 2014). Adaptation to hand-tapping was also selective to external position, not to the adapting hand (Anobile et al., 2016a; Togoli et al., 2021). All this makes good sense: the system wants to know the number of objects in a particular part of the scene, not everywhere, or on a particular part of the retina.

For the most robust case of adaptation, adaptation to spatial arrays, it has been frustratingly difficult to determine if the spatial selectivity is retinotopic or spatiotopic. That the effect is not entirely retinotopic is clear, as it does not work with a static demonstration, of the type typically used for colour and orientation: adapt to a dense dot field on the left, then shift your gaze to a moderate field on the right, and its numerosity is unchanged! But when I tried to demonstrate spatiotopy, first with John then with postdoc David Murphy, the effects were never compelling. A recent study by Myers et al. (2025) suggests an explanation. They found numerosity adaptation to be neither retinotopic nor spatiotopic, but object-based, which makes good ecological sense.

Not only does the number system need to know the number of items in a region, it also needs to know the number of items of a particular type, not of everything. Following up John's and my unpublished pilot studies, Paolo Grasso showed that number adaptation is selective for colour (Grasso et al., 2022): adapting to blue dots reduces the perceived numerosity of blue dots far more than of green dots (see video 3). Similarly, with audio sequences, adapting to high-pitch tones and testing with the same frequency caused twice the magnitude of adaptation as testing with lower-pitch tones. Adaptation selectivity has been demonstrated also for differences in luminance polarity, but not motion direction or letter orientation (Caponi et al., 2025).

The selective nature of visual numerosity adaptation, despite its susceptibility to cross-modal adaption, points to a highly functional mechanism. Adaptation can be complex, as contingency aftereffects such as the McCollough effect (McCollough, 1965) show. But how can numerosity adaptation be so selective to the different colours of visual stimuli, yet generalize across sensory modality, and across space and time? This is a fascinating question meriting much further research. However, it is not unreasonable: while adaptation was strongest for similarly coloured items, it remained significant for unmatched colours (Grasso et al., 2022). The complex phenomenon of numerosity adaptation seems to comprise both selective and non-selective components, revealed by different experimental designs.

Does Adaptation Reflect Real Perceptual Changes?

Given that much of the psychophysical evidence for numerosity mechanisms comes from adaptation studies, it is important to demonstrate that adaptation affects sensory and perceptual processes directly, rather than causing response biases. Adaptation studies necessarily rely on subjective reports of numerosity, which could be influenced by cognitive processes, rather than reflect genuine changes in perception. For example, a consistent bias after adaptation, like ‘if unsure respond “more”’, could in principle affect results in a very similar way to adaptation (Morgan et al., 2012). Many of the effects reported here are so strong and robust that they are easily demonstrated with online videos. Others, however, like the effects of adaptation to action, or ‘reverse adaptation’ (adapt to low numerosities), are more subtle and do not survive a one-shot demonstration in uncontrolled conditions. Yet it is these counterintuitive adaptation effects that most strongly demonstrate the existence of a generalized sense of number. Motivated by the principle that ‘the weight of evidence for an extraordinary claim must be proportioned to its strangeness’ (Laplace, 2012), we obtained more objective evidence for action-based adaption of numerosity.

Maldonado Moscoso et al. (2020) leveraged on a clever technique of Gallagher et al. (2019), based on the principle that observer confidence should be lowest when judgments are most difficult. Adaptation is typically measured by shifts in psychometric functions, the probability of an observer perceiving a ‘test’ stimulus presented to the adapted region to be more numerous than a ‘probe’ stimulus in the unadapted region of the screen (see Figure 3B). The magnitude of adaptation is quantified by the horizontal shift in the point of subjective equality (PSE), corresponding to the numerosity where the curves cross 0.5 probability. Adapting to fast tapping shifts the functions (and the PSEs) to the left, slow tapping to the right (Figure 3B).

Figure 3.

Illustration of the effect of motor adaptation on apparent numerosity of dot clouds. (A) Illustration of the experimental setup. Observers tapped either rapidly or slowly, at a specific location with hand concealed. (B) Fitted psychometric functions, showing the proportion of trials where the test stimulus (at the adaptor site) appeared more numerous than the reference. Slow tapping shifted the psychometric functions leftwards (blue curves), implying that apparent numerosity at the tapped location had increased, fast tapping rightwards. (C) Average confidence in the judgements. The minima in confidence occur at the point of subjective equality (PSE), where the test and reference are perceptually indistinguishable. The minima shift robustly with the two adaptation conditions, implying that adaptation affected the actual perception of numerosity, not just the response. (D) Control condition where the psychometric functions were biased by selective reward, given double points for correctly identifying “less than” in some sessions (blue curves in panel E) and “greater than” in others (red curves). (E) The rewards caused robust and significant shifts in the psychometric functions, similar to the adaptation effects. (F) The robust shifts in PSE were not accompanied by shifts in confidence minima. Adapted with permission from Maldonado Moscoso et al. (2020).

When two numerosities are subjectively similar, judgements should be most difficult, resulting in low confidence and slower responses. Figure 3C shows average confidence ratings, highest when the stimuli were most different, and lowest when they are similar (at the PSE). After adaptation, the minima in the reaction time measures were displaced in the same direction as the PSEs, to around 16 after adapting to fast tapping, and 18 after slow tapping. This clear and significant shift in minima show that the difference was genuinely perceptual, not a response bias. As the pattern of confidence changes with adaptation, it cannot be the source of the bias (‘if unsure say “more”’). Reaction times (not shown, but available in Maldonado Moscoso et al., 2020) were also highest at the adapted PSEs, further proof that they were the most difficult judgements to make. Adaptation must affect sensory or perceptual processes, not just bias responses. The same technique confirms that ‘reverse adaptation’ (adapting to low numerosities) causes real perceptual changes, evidenced by robust shifts in peaks of confidence ratings and reaction times (Benedetto et al., 2026).

On the other hand, cognitive biases do not cause shifts in confidence minima. Maldonado Moscoso et al. (2020) repeated their experiment without adaptation, but biasing responses by doubling the reward for correctly identifying tests ‘lower’ than the probe (and ‘higher’ in other sessions). Figure 3C shows that the manipulation mimicked the response biases and the shifts in PSE caused by adaptation (Figure 3D), but the point of minimal confidence in judgments was unchanged by the reward regime (Figure 3F). This is because the representation of the stimulus was unchanged by the reward regime. If adaptation to tapping affected only the response, like rewards, there would be no shift in confidence. Similarly, other debatable forms of adaptation, like adaptation to ‘implied motion’ (stationary scenes implying motion), cause a clear shift in PSE with no accompanying shift in confidence minima (Gallagher et al., 2021).

These experiments furnish unequivocal evidence that cross-modal and reverse adaptation of numerosity is real, as it would be impossible to fake the orderly changes in confidence and reaction time functions (participants were not even told that reaction times were being recorded, or asked to speed responses). It also shows that number adaptation acts on sensory mechanisms, rather than perceptual decisions or responses.

Further Challenges to Number Adaptation

Recently, the concept of number adaptation has again been challenged by Yousif et al. (2024). Amusingly, the new challenge was not to the ‘novel’ concept of number adaptation, but to the ‘classic theory of number adaptation’. Yousif and colleagues claim that number adaptation does not exist, and all effects are elegantly explained by what they term the ‘old news’ hypothesis. ‘Old news’ is a term coined by Ned Block (2014) to describe one of the functional roles of adaptation, filtering out ‘old news’ to highlight change. Yousif et al. (2024), however, do not use the term as a metaphor, but as an alternate, competing theory.

Interested readers should read their proposal for themselves (Yousif et al., 2024), along with our detailed response (Burr et al., 2025). But at risk at oversimplifying their theory, the ‘old news’ essentially refers to individual dots, rather than numerosity-tuned mechanisms. After adaptation, sensitivity to test dots falling on or near the position of the adapting dots will be reduced (as they are ‘old news’), so the numerosity of the dot field will seem lower.

Fortunately, this theory makes several clear and readily testable predictions. The most obvious (as with the previous challenge) is that cross-modal adaptation should not occur, as there are no dots in the adaptors to become ‘old news’. Similarly, ‘reverse adaptation’ (adapt to low numerosities) cannot be explained, as ‘old news’ can only drive the numerosity down, not up. Yet both cross-modal and reverse-adaptation have been repeatedly reported in many competent laboratories; and importantly, both cause orderly shifts in minimal confidence and maximal reaction times (Benedetto et al., 2026; Maldonado Moscoso et al., 2020; Figure 3), which could not be faked, intentionally or otherwise. These studies show it is the number sense, not the individual speckles on the hen, that adapts.

Two other predictions are interesting. One is that the adaption should be far stronger when the test and adaptor dots are aligned (so they become ‘old news’) than when they are not. We (Burr et al., 2025) measured adaptation in the two conditions, and found it to be equally strong in both cases, about two just noticeable differences (JNDs, or threshold differences), with no measurable difference between them. Yousif et al. (2024) did not measure the total effect of adaptation, but compared the two conditions directly, and reported a slight difference of about 1/5th JND. This difference is almost certainly due to forward masking (Bachmann, 1994). But even if it were not, it could hardly serve as the only evidence for demolishing the age-old concept of adaptation: their theory requires overlap of dots, but when not a single adapting dot coincides with the test (to make it ‘old news’), at very least, 90% of the effect remains. Grasso et al. (2025) tested the prediction even more rigorously, leveraging on the colour specificity of numerosity adaptation, and showed conclusively that the local novelty predictions of the ‘old news’ theory failed completely. Another clear prediction is that moving dots should not adapt, as being in constant motion makes them all ‘new news’: but casual observation of video 4 shows that is not the case.

Strangely, Yousif and Clarke (2025) now agree with us that number is a primary perceptual quality, based on the speed and spontaneity of processing, and the susceptibility to object-based illusions like the connectedness effect. However, while acknowledging number to be a primary perceptual quality, they insist that unlike all other known perceptual qualities (colour, motion, orientation, etc.), number perception is immune to adaptation. Despite the spectacular demonstrations of adaptation, the evidence that it acts on perceptual rather than decisional processes, the direct imaging of number adaptation in humans, the indisputable evidence for cross-modal and reverse adaptation (confirmed by indirect measures of confidence and reaction times), and the abject failure to verify a single prediction of their alternate theory—despite all this rigorous scientific evidence, they maintain that numerosity is uniquely the only perceptual system not equipped with this ubiquitous neural mechanism of self-regulation.

The Functional Role of Number Adaptation

While there can be no reasonable doubt that the number sense is highly susceptible to adaptation, we still lack a firm understanding of its functional role. Perhaps this is not surprising, given that neuroscientists are still discussing the role of sensory adaptation in general (e.g., Kohn, 2007; Webster, 2015). However, number adaptation may be particularly important in understanding the role of adaptation of perceptual and cognitive systems, as it is a fairly simple, one-dimensional quantity that is easy to define, yet linked to more complex functions, like mathematics (Halberda et al., 2008).

The most common explanation for adaptation is selective ‘fatigue’ of neural mechanisms. We know that this occurs for neurones selective to number, as neural adaptation (Grill-Spector & Malach, 2001) underlies one of the major imaging techniques used to study number selective neural mechanisms: repeated presentation of a 16-dot array selectively attenuates the BOLD response to arrays of 16 or similar numbers of dots, but not to more distant numerosities (Piazza et al., 2004). And the BOLD response changes after even short periods of adaptation (Castaldi et al., 2016; Tsouli et al., 2021). However, none of these studies speak to its functional role.

The clearest textbook example of the beneficial effect of adaptation is undoubtedly light adaptation in the retina (Shapley & Enroth-Cugell, 1984), which allows our system to operate with maximum efficiency over changes in light intensity around 10 orders of magnitude. Figure 4 illustrates the effect for a cat ganglion cell, showing response curves to five different background illuminations, varying over five orders of magnitude (Sakmann & Creutzfeldt, 1969). Sensitivity changes multiplicatively with average illumination, so the same cell responds efficiently over the limited range spanning the ambient level, preventing response saturation and balancing sensitivity across mechanisms, thereby maximizing the information they can carry. Without this automatic gain control, we would be effectively blind most of the time.

Figure 4.

Paradigm example of how adaptation increases efficiency by extending the working range of neural units. The curves are responses of a cat ganglion cell in response to stimulation by a small spot against various backgrounds (indicated by the upper numbers, in log cd/m²), as a function of test luminance (abscissa). For each background luminance, over a range of five orders of magnitude, the same retinal cell responds to luminances straddling the background adaptation level. This is the primary mechanism allowing vision to be effective over the entire visible range, spanning about 10 orders of magnitude luminance variation. Reproduced from Sakmann and Creutzfeldt (1969).

As the brain tends to recycle successful strategies, other forms of adaptation, including numerosity, may serve a similar purpose. However, few attributes vary over the range that light levels do, and it has proven difficult to demonstrate impressive improvements in sensitivity in other domains. For example, much research has been aimed at understanding adaptation to faces, with some interesting results, including ‘norm-based’ theories of face perception, which posit that faces are not encoded in absolute terms, but relative to an adaptively average (Leopold et al., 2005; Rhodes & Jeffery, 2006). However, while the theorizing has been very interesting, there has been no clear evidence comparable with that shown in Figure 4.

Several lines of evidence suggest that adaption to number is functional, rather than a simple byproduct of neural fatigue. The effects are very strong, even after very short (250 ms) periods of adaptation (Aagten-Murphy & Burr, 2016), insufficient for effective neural fatigue, suggestive of a very dynamic system. Other evidence is the complex contingency with colour (Grasso et al., 2022), reminiscent of other contingent aftereffects, like the McCullough effect. It seems unlikely that this results from selective ‘fatigue’ of neurones with dual specificity to number and colour (which have never been reported, nor likely to exist). The selectively more probably points to a complex and flexible mechanism serving to maximize efficient and independent estimation of the number of red and green apples on the tree. When contingency effects were fashionable, John Mollon (1974) observed that ‘the number and complexity of contingent aftereffects and their strange persistence suggest that they are in some ways more akin to the phenomena of conditioning than to sensory adaptation’. Barlow (1991) adopted an information-theory approach to suggest that contingent adaptation could serve to decorrelate the selectivity to orientation and colour, maximizing efficiency of the two-dimensional colour-orientation space. Certainly, the complex link between colour and numerosity seems a promising area to search for evidence of adaptation-induced efficiency.

Imaging studies suggest that adaptation may serve to extend the functional range of numerosity mechanisms. Tsouli et al. (2021) showed that adaptation to either one dot or 20 dots changed the tuning of numerosity maps, towards the adaptation numerosity. This in principle could expand or contract the numerosity range in the direction of the prevailing (adapting) numerosity, as well as providing a neural substate for perceptual adaptation.

A related putative role for adaptation is ‘predictive coding’ (Rao & Ballard, 1999; Srinivasan et al., 1982), stemming from the long-standing idea that the brain builds predictions about the world, and encodes only deviations from the predictions, rather than the actual stimuli (Gregory, 1980; Von Helmholtz, 1925). Predictive coding saves metabolic resources and allows the system to use its full capacity to signal only the errors, or unexpected stimuli, and may help make these errors more conspicuous (Barlow, 1990; Stocker & Simoncelli, 2005). Adaptation could participate in the predictive coding by attenuating the predictable component of the stimulus, in the spirit of Block's (2014) metaphor of ‘old news’.

Much recent research has focused on the notion of predictive coding, particularly the research line termed ‘serial dependence’, studying how previous stimuli influence the responses of currently viewed stimuli (for reviews see: Cicchini et al., 2023; Pascucci et al., 2023). Indeed numerosity perception was one of the first perceptual systems to be shown to be susceptible to serial dependence, explaining, amongst other things, the curved numberline without resorting to putative logarithmic processes (Cicchini et al., 2014). However, serial dependence effects are generally attractive, towards the previous stimulus, while adaption is repulsive. The two effects may represent different sides of the same coin, but exactly how they come about is still far from clear, despite much interest in the issue (Chopin & Mamassian, 2012; Stocker & Simoncelli, 2005; Taubert et al., 2016). So while predictive coding remains a plausible function of adaptation, we are still a long way away from full understanding of how it may operate. This remains a very fruitful avenue of future research.

Concluding Remarks

Studying numerosity has been fun. It is the last project I shared with my friend and mentor, John Ross, towards the end of his eighth decade. I continued the research line with several of my own students and postdocs, many of whom are now acknowledged experts in the field. I name in particular the co-editors of this special edition, Elisa Castaldi, Giovanni Anobile and Roberto Arrighi, and also Marco Cicchini. All are pursuing very interesting lines of research, including development, links between numerosity perception and math (including dyscalculia and math-anxiety), brain-imaging, pupillometry, and a fascinating new theory linking action, numerosity perception and math development (Anobile et al., 2021).

While John's and my strong claim was that numerosity should be considered a perceptual system, it is also obvious that it borders closely with conceptual systems, particularly math cognition, which makes it particularly interesting. There is a clear link between math achievement and numerosity perception (but not texture perception) (Anobile et al., 2016b; Halberda et al., 2008), strongly implicating numerosity estimation in math acquisition. Math strategies can be used to group arrays into subitizable chunks, improving discrimination by extending the subitizing range (Anobile et al., 2020; Maldonado Moscoso et al., 2022; Starkey & McCandliss, 2014). The strong links with math, and the fact that it is easily defined and quantifiable, make numerosity a particularly fascinating area of study, with many ramifications. I look forward to seeing how the field progresses over the next few years.