Texture density discrimination is more precise than number discrimination

Abstract

Density information is a possible primitive for the perception of numerosity. It has been argued, however, that the perception of numerosity is more precise than density perception at low numbers, whereas density is more precise for high numbers. An interpretive problem with the stimuli used to make those claims is that actual stimulus density was often mis-specified owing to an ambiguity regarding the idealized versus actual filled area. This ambiguity had the effect of underestimating density precision at low numerosities. Here we used a novel method of stimulus generation that allows us to accurately specify stimulus density independent of patch size and number, while varying patch size from trial to trial to dissociate numerosity and density. For both numerosity discrimination and density discrimination, we presented single stimuli in central vision for comparison with an internal standard. Feedback was given after each judgment. Using well-defined densities, density discrimination was more precise than numerosity perception at all densities and showed no evidence of varying as a function of density, as previously hypothesized. This was found with 8 practiced observers, and then replicated in a pre-registered study with 32 observers. As expected, feedback nullified size biases on number judgments, showing that observers were adaptively combining density and size. Reanalysis of data from a recent investigation of downward sloping Weber fractions for numerosity showed that the square root–like effects in those sorts of studies were most likely owing to reductions in patch size variance that were correlated with increases in density.

Introduction

Discrimination data in the perception of number and density suggest that these dimensions might be linked: Explicit comparisons of perceived density and numerosity are both biased by the size of the texture patches (Dakin, Tibber, Greenwood, Kingdom, & Morgan, 2011). Likewise, separate aftereffects of density and size can both alter perceived numerosity (Durgin, 1995; Durgin & Martinez, 2024; Zimmermann & Fink, 2016). Mathematically, it may appear most obvious that density is the ratio of number and area. However, by simple algebra, numerosity could be calculated as the product of density and area. This second scheme is supported by the fact that density appears to be a local feature that could be extracted early in visual processing via divisive contrast normalization, thus reflecting known mechanisms of early vision (Durgin, 2001; Park & Huber, 2022). Such a sensory representation of density could support the estimation of visual number, as well as providing information about spatial properties, such as surface layout and local surface orientation (Knill, 1998; Gibson, 1986).

The psychophysics of numerosity perception does not imply that number is sensed directly

It has long been acknowledged that humans and many non-human animals demonstrate numerosity judgments that are well-behaved. This capacity has been dubbed the “number sense” (Dehaene, 1997) and attributed to a system sometimes described as the approximate number system (ANS; Feigenson, Dehaene, & Spelke, 2004). Explicit estimates of numerosity have been studied in humans for more than100 years, and traditionally divided into a subitizing region and an estimation region (Kaufmann, Lord, Reese, & Volkmann,1949). More recently a second division in number estimation of spatial number (dot patches) has been proposed between numbers below and above 20 dots (Portley & Durgin, 2019). Whereas subitizing performance is thought to be accurate (essentially errorless) up to about four items for two-dimensional spatial number and three for one-dimensional spatial number (Atkinson, Campbell, & Francis, 1976), it has long been known that grouping of subitizable units (e.g., a line of dots split into two colors) can extend subitizing-like behavior (Atkinson, Francis, & Campbell, 1976). However, the estimation range beyond the subitizing range has sometime been regarded monolithically. Portley and Durgin (2019) found instead that that, in log space, there was a sharp elbow between a linear range up to 20 elements (where mean number estimates were accurate without any training), and for numerosities beyond 20. Above 20, numerosity estimates, although linear in log–log space, have a log–log slope of substantially less than 1, meaning that the underestimation becomes proportionally greater as number increases. Portley and Durgin called this the superdigital range (because of it being clearly analog) and suggested that accurate performance up to 20 might be due to estimation based on a small number of subitizable subsets.

Durgin and Portley (2023) found that this elbow seemed to be an intrinsic limit, independent of whether the displays to be estimated were presented for 0.5 seconds or 2.5 seconds. However, the limit of 20 has turned out to be specific to planar arrays. Durgin, Aubry, Balisanyuka-Smith, and Yavuz (2022) observed that when estimating the numerosity of stochastic temporal events with mean presentation rates too rapid for counting, the elbow occurs at about four or five elements, and underestimation (linear in log–log space) occurs thereafter (Durgin et al., 2022; Whalen, Gallistel, & Gelman, 1999).

The point here is that some number tasks (that are accurate without training) appear to be governed by cognitive strategies which operate only over a limited range. These tasks are therefore probably uninformative about the ANS more generally. For example, a recent study (Sanford & Halberda, 2024) that looked as numeric estimation of fairly small numbers (up to 10) in naturalistic images, concluded that their results could not be explained by low-level features (although there was a strong correlation with density). However, results based on estimation data for numbers in this range may not really bear on the ANS or numerosity perception more generally (Gordon, 2004) and studies seeking to look at the ANS would do well to look at superdigital ranges or use different kinds of task. Indeed, a hallmark of the ANS is not only that it is approximate, but that these approximations tend to underestimate (Krueger, 1972; Krueger, 1984; Whalen et al., 1999). Calibration training can make numeric estimates linear over a larger range (e.g., Izard & Dehaene, 2008), but this is true for all psychophysical dimensions (Stevens, 1956), and does not really imply that the underlying system is not log coded and thus analog.

In tasks that do not involve numeric estimation, such as the comparison of two simultaneously presented sets of dots, it is not yet clear whether similar kinds of specialized strategies are used for smaller numbers. However, it remains probable that a variety of different kinds of visual information may contribute to judgments concerning the numerosity of spatial arrays of elements (Gebuis, Kadosh, & Gever, 2016; Henik, Gliksman, Kallai, & Leibovich, 2017; Lourenco & Aulet, 2023; Negen, 2025). Although there are special cases that suggest that visual cognition of numerosity may have multiple pathways, texture density, which is coded early in the visual stream, has been deeply implicated in the perception of numerosity (Miller, 1956; Dakin et al., 2011).

The likely role of element density in number discrimination judgments (or tasks)

Human perceptual judgments are the output of a complex, multi-stage processing system, with multiple stages of visual encoding, and flexible circuits for “reading out” relevant information from the encoding stages. With respect to visual encoding, texture-based models of density representation (e.g., Dakin et al., 2011; Morgan, Raphael, Tibber, & Dakin, 2014) are the most detailed and neurally plausible theories available. These models suggest that the early visual system (with relatively small receptive fields, and known spatial filtering properties) encodes information that can be read out via a simple computation to provide an early, local density estimate (e.g., via the comparison of neural populations tuned to different spatial frequencies, as in Dakin et al., 2011). For high numerosities, at least, adaptation to dense texture produces aftereffects on perceived number that are retinotopic (Durgin & Proffitt, 1996; Zimmermann, 2018), consistent with the presumption that such adaptation may be at an early site that represents density. Huk and Durgin (1996; see Durgin, 2016) showed that number estimation is likewise influenced by this kind of adaptation. Like Durgin (1995), they observed weaker adaptation for numbers less than 20 than for those above 20.

Of course, density must be integrated over area to produce estimates of numerosity. Zimmermann and Fink (2016) observed that adaptation to large or small discs (i.e., large or small relative to the patch size to be later tested), produced size aftereffects on dot patches, and produced consistent shifts in perceived numerosity, but not in perceived density. This observation is consistent with the idea that patch size and patch density are each used in evaluating spatial numerosity, and that numerosity was affected by size adaptation for this reason. Zimmermann and Fink observed these size effects on numerosities as low as 13, presented about 10° in the periphery.

In a more recent study, Durgin and Martinez (2024) measured size and numerosity aftereffects following adaptation to large or small patches of dots of high or low density. They showed that this method produced systematic numerosity aftereffects as well as explicit size aftereffects, even when adapter numerosity was controlled for. By computing the implicit shift in perceived density that would account for the numerosity aftereffects observed, after accounting for size aftereffects, they found that density adaptation was fairly symmetric: Adaptation to low densities made middling densities appear denser, and adaptation to high densities made middling densities appear less dense. This observation replicates other demonstrations of density aftereffects (e.g., Sun, Kingdom, & Baker, 2017), and extends to cases where patch size varied from trial to trial to clearly dissociate density from numerosity.

Disputes over the nature of numerosity aftereffects are parsimoniously resolved by defaulting to density-based visual encoding

Durgin (1995; Durgin & Huk, 1997; Durgin & Proffitt, 1996) showed that perceived numerosity was affected by density adaptation, and that this was not reducible to spatial frequency adaptation (Anstis, 1974; MacKay, 1973). However, Burr and Ross (2008), while replicating the observation of Durgin, argued that they had identified an entirely new form of adaptation that was evidence of a “visual sense of number” being adapted. Novel to their treatment (at that time) was the observation that perceived number could be increased by adaptation to a small number value (cf. Yousif, Clarke, & Brannon, 2024). However, missing from their treatment was any control for density; that is, they did not distinguish density from numerosity by varying patch size. Instead, Durgin (2008) immediately showed that adaptation to greater local density rather than greater size or number produced stronger reductions in perceived number, suggesting that density representation, not number representation, was the primary site of adaptation.

In claiming that perceived number was adapted directly, Burr and Ross (2008) argued that texture density was not the affected dimension because aftereffects that they attributed to number were not specific to texture (similar to the observations of Durgin & Huk, 1997). However, as noted by Yousif et al. (2024), later studies have claimed that number adaptation does depend on early features, such as the colors involved (Grasso, Anobile, Arrighi, Burr, & Cicchini, 2022). These effects seem consistent with low-level interpretations and replicate prior observations that density aftereffects can be made contingent on color information, for example (Durgin, 1996), but they appear inconsistent with the argument that spatial number adaptation is not related to texture properties (see also Durgin & Proffitt, 1996).

As reviewed, studies of visual adaptation tend to support the view that visual number beyond about 12 to 20 dots may be derived from patch density and patch size (Durgin, 2008; Durgin & Martinez, 2024; Zimmermann, 2018; Zimmermann & Finke, 2016). For example, Durgin (2008) showed that a reduction in perceived numerosity could be achieved by adapting to a relatively dense, yet smaller and less numerous, dot patch. Conversely, Durgin and Martinez showed that an increase in perceived numerosity could be achieved by adapting to a relatively sparse, yet larger and more numerous, dot patch. These two findings, together, implicate density as an adaptable dimension that (along with patch size adaptation) could underly most numerosity aftereffects.

Although one recent study claimed to show that adaptation was to number rather than density (DeSimone, Kim, & Murray, 2020), that study confounded size adaptation with number adaptation. Instead, when aftereffects of perceived size, density and numerosity were all measured simultaneously (in a replicating extension of this work), adaptation of density and size were found to be sufficient to fully account for numerosity aftereffects (Durgin, Love, & Taylor, 2025), with no additional contribution from number. In particular, numerosity aftereffects were strongly correlated with size aftereffects and with density aftereffects, but density aftereffects were not correlated with size aftereffects. Thus, despite numerous claims that number adaptation cannot be based on density adaptation (e.g., Burr & Ross, 2008), in the few studies that have directly measured size and number aftereffects simultaneously, the evidence is fairly overwhelming that much of what is called number adaptation is likely due to a combination of patch-size adaptation (Zimmermann & Fink, 2016) and the adaptation of well-understood early visual processing stages that might best be understood as measuring local density (Durgin & Martinez, 2024). In general, adaptation studies that consider size, density, and number independently find evidence that adaptations of perceived density and perceived size, rather than number, are what primarily produce numerosity aftereffects.

Cross-modal “numerosity” adaptation is uncompelling

As evidence for the number adaptation view, several studies have reported intermodal adaptation effects using temporal number (e.g., Arrighi, Togoli, & Burr, 2014). However, analogous to the concerns described about differentiating number and density, these studies have generally been designed in a manner that conflates number adaptation and frequency adaptation. For example, Arrighi et al. used adapters of 2 or 8 Hz of fixed durations (6 seconds) and then used temporal test numbers of fixed duration (2 seconds) such that frequency and number were confounded in their test stimuli. This means that the difference between adapters and testers was in frequency as well as number. However, the data seem to suggest that frequency might be the adapted variable. Note that the 6-second, 2-Hz adapter used in this study contained 12 events yet was reported to increase the perceived numerosity on all test stimuli from 5 to 15 (including, for example, test stimuli of 10 events—presented at ∼5 Hz over 2 seconds). This increase makes perfect sense as temporal frequency adaptation, if you think of the adapter as 2 Hz and the test as 5 Hz, whereas it seems to make little sense to conclude that a 12-event adapter should make a 10-event test appear more numerous because of number adaptation. That said, Yousif et al. (2024), using open science methods, reported that they could not replicate the phenomenon, weakening the relevance of multi-modal studies to our focus on visual density and numerosity.

Arrighi et al. (2014) also reported cross-format adaptation of spatial number by visual temporal number that would be hard to understand in terms of frequency adaptation. However, the complexity of the paradigm used to establish this effect, the lack of symmetry in the magnitude of the “adaptation” effects reported between temporal and spatial number, the repeated use of the same small set of participants across multiple paradigms in a magnitude estimation procedure, and the absence of any mechanism to protect against selective publication does not render this surprising observation a strong basis for scientific theorizing—especially given the difficulty Yousif et al. (2024) had in observing even the cross-modal effect in the temporal domain. Moreover, the possibility of contamination by generalized magnitude biases, attentional biases related to task-switching effects, size adaptation to the annular displays, or other high- or low-level effects, were not considered in the paper. Although a later study using fast-tapping as the adapter sought to rule out decision biases (Maldonado Moscoco, Cicchini, Arrighi, & Burr, 2020), no alternative low-level hypotheses were considered (e.g., effects of task-switching in a sequential measurement paradigm), while the magnitude of the reported effect (12%) was well below a perceptual just noticeable difference, and thus could account for only a sliver of the effect on perceived numerosity that density adaptation has been shown to produce.

Summary of conclusions regarding purported number adaptation effects

Nothing in this discussion denies that numerosity could be a well-behaved psychophysical dimension. Nor are we suggesting that number adaptation, per se, could never happen. But we do note that, in psychophysical studies that have sought to dissociate number adaptation from density and size adaptation, the very large biases observed in spatial number perception after adaptation seem to be determined primarily by density adaptation and secondarily by size adaptation, with very little left for number adaptation to explain. In the case of temporal number, the proper tests have simply not been done.

Ambiguities in prior attempts to compare the precision of texture and numerosity judgments

Anobile, Cicchini, and Burr (2014) proposed that texture density and number can be distinguished based on the idea that density discrimination becomes more precise (approximately as a square root function) as density increases, whereas number discrimination is constant. Their initial evidence was based on a steeply declining Weber fraction (beyond a certain density) in a number-discrimination task when patch size was held constant. They attributed this to a switch from number to density discrimination. Anobile et al. provided only two data points concerning actual density discrimination. Their argument depended, instead, primarily on the claim that density discrimination was equivalent to discrimination of interdot distances and thus should follow a square root. In a subsequent paper, Anobile, Turi, Cicchini, and Burr (2015) reported that the downward sloping Weber fractions could be aligned on a metric based on averaging the distances of each dot to its nearest neighbor, concluding that this meant that the effect was due to crowding-like mechanisms associated with specific levels of texture density.

These conclusions may have been premature. Indeed, the clearest evidence that the downward-sloping functions observed by Anobile et al. (2014) are not related to increases in retinal texture density was published 30 years previously, by Burgess and Barlow (1983). Burgess and Barlow had observed this surprising behavior of increasingly lower Weber fractions for higher numerosities. They specifically tested the hypothesis that it was due to density by varying the viewing distance to their displays so as to increase or decrease the retinal density of the displays. They found no significant alterations in performance over a range of distances representing changes in scale by up to 1 to 16. They, therefore, concluded that the effect was not due to density. Note that their dot fields were not spatially separated, but appeared adjacent to one another on either half of the display. More recently, Durgin (2025) replicated the scaling manipulation tested by Burgess and Barlow (without varying viewing distance), using separated stimuli like those used by Anobile et al. (2015). Like those of Burgess and Barlow, these new data showed that the downward sloping discrimination function was independent of retinal stimulus density. (Note that most data show a slope between 0.2 and 0.4 rather than 0.5.)

So far, demonstrations of these downward sloping Weber fractions have been limited to circumstances in which the notional (i.e., possible) regions in which dots were thrown was held constant. As noted by Durgin (2025), this introduces an alternative explanation of the reduction in decision variance implied by the declining Weber fractions. As numerosity (and thus notional density) increases, the notional area where dots are randomly allowed to appear will become more full. More specifically, the higher the number of dots, the lower the variance in the size of the area within the convex hulls of the patches of dots (the “convex hull” can be thought of intuitively as a rubber band stretched around the dots). Because Anobile et al. (2015) preserved the same minimum interdot distances in their displays when testing the dot-distance hypothesis, their nearest neighbor metric was completely confounded with the area variance explanation proposed by Durgin. In fact, Durgin showed that the expected variance in the convex hull of the dot patch decreases dramatically as numerosity increases (with a slope of approximately −1 in log–log space.) Thus, the decreasing Weber fractions in numerosity judgments might be only apparently related to density, but actually owing to concomitant reduction in the variance in area (convex hull) information. This factor could explain why the effects are identical at different scales.

The neglect of consideration of actual area information is a deep problem in the work of those who have argued that number is sensed directly. For example, in the study of Cicchini, Anobile, and Burr (2016) that sought to pit area and density against numerosity, they defined both density and area notionally, as if the area used to constrain the dot patches were full. Given the extremely high variance in actual area for lower numerosities (as measured by convex hulls), there would also have been extremely high variance in empirical mean density as well: If 15 dots are randomly squeezed into a small convex hull or spread over a large one, they are the same number, but are they the same in size or in density? In the analyses presented by Cicchini et al. (and by Anobile et al., 2014), the answer was assumed to be yes. Density was defined as the number of dots divided by the notional area, and area was defined as the notional area rather than the objectively filled area. This factor could explain why their findings indicated that participants had higher Weber fractions (poorer discrimination) for density than for numerosity at lower numerosities. Numerosity was the only one of the three stimulus variables in the study that was accurately represented in their analyses.

In general, the idea that the precision of density discrimination is worse than that of numerosity (at low numerosities) likely results from artifacts involving the mis-specification of actual stimulus density. This mis-specification means that, when participants may have correctly identified an empirically smaller patch of 10 dots as denser than an empirically much larger patch of 12 dots, this would have been treated (by Anobile et al., 2014 and by Cicchini et al., 2016) as an error in their judgment, rather than an error in the stimulus definition (Figure 1). Clearly, estimates of the precision of density discrimination in such studies would have grossly underestimated sensitivity to differences in actual (presented) density. Note that, in the data of Cicchini et al., estimated density discrimination was superior to (more precise than) numerosity discrimination for most numerosities; the exception was the low numerosities, where substantial variance in empirical stimulus density was unaccounted for. This ambiguity regarding area means that the apparent superiority of numerosity discrimination at low numerosities may have been artifactual, although the effects of misdefining area may be only part of the story.

Figure 1.

Illustration of an ambiguity in prior studies of density discrimination. Which patch of dots is denser? Although the equation for density seems simple (i.e., number divided by area), the definition of “area” in such random-dot stimuli actually requires careful consideration. Consider this example: There are 10 dots on the left and 12 on the right. The notional area (a common definition assumed in prior work) is indicated by the lighter circular area. The convex hulls (which we argue are a more perceptually relevant definition) are depicted by the line drawn around the dots. In this example, the area containing the dots (defined by the convex hull) is half as large on the left as on the right, meaning that the left patch is, arguably, 67% denser than the right patch. In studies purporting to show that density discrimination is worse than number discrimination at low numerosities (e.g., Anobile et al., 2014; Cicchini et al., 2016), the notional density difference was defined according to the notional area (i.e., density was defined as 20% higher on the right). Therefore, choosing the left patch as the denser one would be considered an error, contributing to the very high Weber fractions for density reported in those studies. Ten thousand simulations using similar randomization parameters, but defining area by means of the convex hull, found that, for a numeric difference of 10 vs. 12 dots, the patch with the smaller number of dots would actually have been denser 38% of the time (and at least 20% denser 18% of the time). Additional ambiguities arise when local clumps of high and low density within a single patch are considered.

In addition to the observation that the variance in actual filled area is negatively correlated with numerosity/density, there is one additional alternative explanation of why increasing the number of dots could decrease the variance of density (and hence numerosity) estimation. There is good reason to believe that perceptual comparisons of numerosity (or mean density) uses only about half the information available (Barlow, 1978; Solomon & Morgan, 2018). The central limit theorem tells us that the variance in the estimate of the mean of a population depends on sample size. In this sense, if area is fixed, then the variance at estimating density should decrease as the number of elements increases (in proportion to the square root of N). So, this is another hypothesis that could help to account for the declining Weber fractions observed as number/density increases in a fixed area. This interpretation would also account for the null effect of scaling.

Note that, for numerosity, other kinds of information, in addition to density information, may come into play at low numerosities, whereas information to specify mean density itself (for late decision processes) may indeed become less certain at low numerosities, given the exigencies of sampling and the central limit theorem: Thus, it seems likely that using densities that are relatively highly variable over space at low numerosities will make it hard for the post-sensory decisions stages to know which information to pay attention to for the sake of performing a psychophysical task regarding relative density. But if integration processes involving early density information feed into processes for evaluating numerosity, it is not unreasonable to conclude that even for stimuli where density is harder to identify at higher levels of processing (late decision stages), number processing may yet be said to have benefited from the use of early mechanisms sensitive to density.

The importance of using well-defined densities

As mentioned, prior studies comparing discrimination performance for number and for density have tended to use textures with random element locations, resulting in naturally high variance of density over space (Anobile et al., 2014; Dakin et al., 2011; Morgan et al., 2014; Raphael & Morgan, 2016; Ross & Burr, 2010), and (as discussed at length earlier), at lower numerosities, uncontrolled variance in actual patch size. Some studies have even defined texture density in terms of the overall proportion of light or dark pixels (e.g., Ross & Burr, 2010, Experiment 2, which claimed, on this basis, that perceived density was affected by luminance, whereas number—measured with completely different stimuli—was not). If density is an early local visual dimension, attempts to measure the limits of density discrimination using high variance textures, and without taking true patch size into account might be somewhat ineffective—simply because the local densities would not have a common value across different regions of the texture. What does it mean to say that one patch of dots is, on average, denser than another, when both are visibly highly variable in density? Judging average density seems like a fairly cognitive task. So, to measure the precision of local density discrimination while minimizing such cognitive computations, stimuli manipulating density should be fairly uniform over space, so that density is well-defined. Such refined control of stimulus parameters should constrain the number of strategies associated with reading out the visual information relevant to perform judgments of density and/or numerosity.

Along these lines, some modern studies of successive and simultaneous contrast in the perception of texture density (e.g., Sun, Baker, & and Kingdom, 2016; Sun et al., 2017) have intentionally used low variance textures to more precisely manipulate and/or measure density locally. Using low variance textures has helped to clarify that density, not numerosity, is the primary adapting dimension (Durgin & Martinez, 2024). In these cases, however, the creation of low variance textures was accomplished by forcing randomly scattered dots to have a minimum interdot distance that is proportional to the square root of the density of the dots (see Durgin, 1995). Although this method can effectively reduce the variance of density, it still depends on defining stimulus density in terms of area (often defined as the area permitted to have dots) and number.

For the present study, we developed an alternative algorithmic approach that permits the (essentially instantaneous) generation of textures that have well-defined, low-variance density independent of their shape, size, and number. This was done by randomly positioning a dot in each cell of a hexagonal grid, where the cell-size of the grid defines the density. Burgess and Barlow (1983) found that numerosity discrimination was improved when they used a similar smooth texture manipulation based on a randomizing dot locations in a square grid. This improvement is hardly surprising, but we emphasize it here to clarify that the use of low variance texture does not disfavor numerosity judgments.

Finally, we note that psychophysical studies of texture regularity (Morgan, Mareschal, Chubb, & Solomon, 2012) have not suggested that semi-regular patterns are processed in a unique way from fully random ones (i.e., they follow Weber's law with respect to inter-element spacing). Variation in element size and orientation can additionally contribute to the perception of relative regularity (Sun, St-Amand, Baker, & Kingdom, 2021), but the present manipulation does not involve such variation. The improvement afforded by defining density in a meaningful way, as well as evidence that low variance of density is not problematic, further validates the approach to stimulus generation put forth in this study.

Measuring precision using an internal standard

Our chief interest in the present study was the precision of judgments (rather than their accuracy), and we sought to use single stimuli on each trial so that the properties of that stimulus would alone be responsible for the judgment. We varied stimulus diameter up or down randomly by up to 20% (relative to the presented standard) from trial to trial so that judgments of numerosity and of density would be clearly dissociated in practice. Because only a single stimulus was presented in each trial, to be compared with an internal standard, we provided feedback on each trial to avoid calibration drift in memory.

Sensory information and decision structures are both relevant to the number–density debate

Our psychophysical measures reflect the output of both sensory processing and later decision stages. Providing feedback, as we did here, could theoretically alter the decision process. This factor is not problematic for our main goal, because feedback should not bias behavior in favor of density. However, there is an interesting asymmetry that arises in the present investigation. By hypothesis, estimates of numerosity must take both density and size into account. Because we varied size from trial to trial, participants could not base their numerosity judgments on a specific density, but they might (over time) get feedback about various combinations of size and density, given that actual area could vary by a factor of nearly 2. In other words, feedback in the numerosity task could help participants to overcome known biases in number perception (Dakin et al., 2011).

Conversely, it might not be obvious to participants that density judgments should be affected by patch size. In fact, Dakin et al. (2011) observed very strong effects of patch size on judgments of density. In principle, participants receiving feedback should be able to notice this if they thought that patch size was relevant to density perception. Based on the fact that both density and size vary with viewing distance, and thus frequently covary with each other, it seems likely that, at the level of decision readout, perceived density could be greatly distorted by size constancy mechanisms. However, it also seems likely that feedback would not alert participants to this association because they are responding to a perceptual variable that seems (to them) to be unrelated to patch size. That is, it might be argued that the participants should be more likely to be able to eliminate patch size effects for a dimension they regard as cognitive (numerosity), than for a dimension the participants may consider perceptual (density).

Experiment 1: Density and numerosity discrimination

We first sought to compare the precision of density and numerosity judgments using improved stimuli that could yield a more definitive answer than poorly defined stimuli used in prior studies. All displays used textures in which density was well-defined locally. Patch size varied from trial to trial so that number and density judgments were clearly differentiated. Numerosity judgments and density judgments were done in separate sessions.

Methods

Open science

This experiment was not registered. All raw data files and a data key file are available on the open science framework (OSF) accessible at https://doi.org/10.17605/OSF.IO/7Z6XH.

Participants

Two authors contributed data, as well as six participants who were unaware of the hypotheses and theoretical goal of the study (mean age, 25 years). All observers had approximately 20/20 vision, using glasses or contacts for correction as needed.

Design

Each participant completed a density session and a numerosity session on separate days, with one-half doing density first and one-half numerosity first. In each session, a unique, pre-randomized testing order was used to measure perceptual precision at each of 9 values of numerosity or of density that encompassed a logarithmic range representing a factor of 16. For number, these values ranged from 20 to 320 dots. For density, they ranged from 1 to 16 dots/deg².

Measurement was done by presentation of each individual stimulus briefly, for comparison with an internal standard. The internal standard for each measurement was initially established using a single, freely viewed standard numerosity or density patch that was presented at the beginning of the measurement block. For numerosity, the standard patches all had the same notional size, but differed in density and number. For density, the standards all had the same notional number of dots, but differed in density and size. Thus, the standards across the two tasks were matched only on density. Feedback was given on each trial throughout the experimental block so as to maintain calibration of the internal standard. Four randomly interleaved up/down staircases with large initial deviations (ranging from 0.54 to 1.88 of the standard), large logarithmic step sizes (1.04⁴), and a combined sampling resolution of 1.04 were used to select stimuli along the dimension of interest with 25 trials selected from each staircase (100 trials total) for each measurement. (For trials with the 20-dot standard, the base factor used was 1.054 instead of 1.040 to address effects of rounding at this level of numerosity.) Because patch size was varied randomly within each measurement block, the area inside the convex hull of each set of displayed dots was recorded (in pixels) for later analyses. For the main analyses, individual psychometric fits were made in log (number or density) space using logistic regression, with just noticeable differences defined as the difference between log values (the difference between the modeled 50% and 75% probability “more” response points for the fit). Thus, the just noticeable difference represented the log of the sum of the Weber fraction and 1.

Apparatus

Displays were presented on an iMac with a resolution of 105 ppd at the viewing distance of 70 cm. Custom software programmed in MATLAB and PsychToolbox (Kleiner, Brainard, & Pelli, 2007) controlled the displays and the collection of data from participants.

Dot patch specifications

To generate dots patterns of a known density, a large, completely regular hexagonal array of dots positions (specified in radial coordinates) was scaled to the intended density and randomly rotated. The resulting array was converted to Cartesian coordinates with an added common random displacement of the array in x and y. Finally, each dot position was radially jittered within its cell of the array by an amount that ensured that dots would be separated from each other by a space of at least half the diameter of a dot. The radial coordinate of each dot was recomputed to be used for truncating the display to a circular radius. The underlying hexagonal structure of the array was not perceptually discernable in the resulting patches. (Fourier analysis confirmed that there was not substantial added power at frequencies corresponding with the hexagonal array; the remaining variance of dot positions thus yielded power spectra that would be expected given the size of the dots themselves.) Example standard patches are reproduced in Figure 2.

Figure 2.

Schematics of stimuli. Images illustrating sample textures corresponding with three of the nine standards (1, 4, and 16 dots/deg²) used for the density task (top three) and all of the nine standards used the numerosity task. Numerosity standards for 20, 40, 80, 160, and 320 dots are shown in the middle row, and for 28, 57, 113, and 226 dots in the bottom row. A single, variable-sized image was shown on each trial. Numerosity test images were 5° ± 1° in diameter. Density standards ranged from 17° in diameter (1 dots/deg²) to 4° in diameter (16 dots/deg²), with test stimuli for each density varying not only in density, but also in diameter (by up to ±18%). Screen resolution was 105 pixels per degree in Experiment 1.

For the numerosity task, the theoretical median size of each dot patch was always 5° in diameter, such that 20 dots required a density of 1 dots/deg² and 320 dots required 16 dots/deg². From trial to trial, the stipulated radius of the circle used varied randomly up or down by up to 20%, which meant that the density of the field was selected so as to produce the required number in the designated area. After dot randomization, the true radius that contained exactly the number specified was used to truncate the coordinate list. The size of the dots used for each standard number value was inversely proportional to the square root of the standard number (i.e., a diameter of 16 pixels or 0.15° for 20 dots but of just 4 pixels or 0.038° for 320 dots). The same set of dot sizes was used for the corresponding densities.

For the density task, the size of the standard display (and thus the median notional size of the test displays) was selected so that it would typically contain 226 dots for the target density (to maintain a consistent amount of information available to specify density), and thus ranged from 4.2° in diameter (for 16 dots/deg²) to 17° in diameter (for 1 dots/deg²). The diameter of the presented patch on any given trial varied randomly up or down by up to 18% from that of the standard for that density. Dakin et al. (2011) found that perceived density was strongly affected by patch size, so the trial-to-trial variation in density had the potential of adding additional variance to the density judgements, but was deemed important to avoid confounding density and numerosity. Conversely, there was no reason to present fewer dots, on average, when the standard density was low. At the highest density, the density patches were smaller than the numerosity patches.

For both number and density tasks, the luminance of the dots with respect to the gray background was the same across all dots within each display. But the luminance used for the dots was varied randomly from trial to trial in a range producing Weber contrasts of 2.1 to 4.2 relative to the background gray (10.6 cd/m²) so that luminance and contrast were relatively uninformative about number or density.

Each test patch was presented for 0.3 seconds. After a key press response indicating whether the test stimuli was higher or lower (in the relevant dimension, either density or numerosity) than the internal standard, a beep indicated an error if one was made, and a variable delay of at least 1 second preceded the following trial, during which a fixation mark was present on the screen. The fixation mark was then erased, and the screen was blank for a variable interval (0.31 to 0.64 seconds) before the onset of the next stimulus patch. Participants were instructed to maintain their gaze at the center of the screen throughout.

Results

Density discrimination judgments were found to be more precise than numerosity judgments. Logistic psychometric functions were fit to the data (in log units) from each measurement block. The mean Weber fractions derived from the data are shown in the left panel of Figure 3. A repeated-measures analysis of variance (ANOVA) on the log Weber fractions confirmed that discrimination was more precise for density than for numerosity, F(1,7) = 8.63, p = 0.02, η² = 0.35. The overall effect size measured across participants was large (D = 1.04). Thus, when both number and density are well-defined, density judgments are more precise than number judgments. The best fitting lines to the data also are shown in the left panel of Figure 3. Both for number and for density, the mean slope was not different from zero. Neither dimension showed any evidence of a downward function.

Figure 3.

Results of Experiment 1. (Left) Mean Weber fractions (log space) for numerosity and density are shown with standard error bars (across participants) as a function of the dot density (expressed as number of dots in 20 deg²) or the numerosity of the standards (for numerosity tasks, the standard patch for numerosity was 20 deg² in area, so the number of dots in the standard was equal to the indicated standard density value). Blue, numerosity discrimination; green, density discrimination. For each task, the dashed line representing the best linear fit is shown as a function of density/numerosity. Error bars represent standard error computed across participants. (Right) Size-based changes in PSE (expressed as percent) for density (green) and for numerosity (blue), based on splitting the data by the median patch size (area within the convex hull) shown on each trial. Error bars represent standard error computed across participants.

Perceived density was affected by patch size

Because patch size varied randomly from trial to trial, we split the data from each measurement by the median patch size (area inside the convex hull of the patch), so that the points of subjective equality (PSEs) could be computed separately for larger and smaller patches for each participant. To do this, we used the same algorithm that was previously used to establish the Weber fractions, but now based on 50 trials each (i.e., separately for patches larger or smaller than the median) from the 100 trials from the experiment. The right panel of Figure 3 show the mean difference in PSEs (expressed as a percentage) resulting from splitting the data by size.

Perceived density was significantly higher for larger patches than for smaller patches. This was tested by an ANOVA on the density PSEs from the size-split data, with patch size (large or small) and standard density as predictors. PSEs were lower for large patches than for small patches, F(1,7) = 156.2, p < 0.0001, η² = 0.91. A lower PSE indicates that large test patches of the same density appeared to be denser than small text patches, and thus a less dense texture was sufficient to seem equal to the standard. Large patches were, on average, 45% larger in area than small patches; the overall magnitude of difference in perceived density was by 18%.

Perceived numerosity was not affected by patch size

The same form of ANOVA on the PSEs for numerosity found that PSEs for numerosity were no different for larger patches than for smaller patches. F(1,7) = 0.01, p = 0.94, η² = 0.003. A Bayesian ANOVA using default priors in BayesFactor (Rouder, Morey, Speckman, & Province, 2012) indicated that the odds favoring the null hypothesis that patch size had no effect were 5.5:1.

No effects of dot luminance

Similar analyses were carried out to test for effects of dot luminance. Neither density nor numerosity comparisons showed any systematic effects of using brighter vs dimmer dots (the mean PSEs differed by less than 1%).

Discussion

The main findings of Experiment 1 were that, with respect to precision, 1) density discrimination was consistently more precise than numerosity discrimination (despite patch size varying from trial to trial), and 2) Weber fractions for density were constant over a 16-fold change in density, showing no evidence of a square root function decrease. With respect to bias, we found that 3) large patches were judged denser than smaller patches, even when only a single patch was viewed on each trial, and despite continuous feedback.

In the absence of feedback, size effects are typically found for numerosity perception even for single trial presentations (e.g., Kreuger, 1972; see also Aulet & Lourenco, 2021; Negen, 2025). The absence of an effect of patch size on numerosity in the current paradigm might be surprising, if not for the expectation that feedback would mute these effects by allowing participants to compensate for noticed biases in the (presumably more cognitive) numerosity task. Conversely, observers may have remained susceptible to size effects on perceived density because they did not consider that size was relevant to density perception. Weber fractions for discriminating density were nonetheless lower than those for numerosity, despite the added variance in perceived density associated with random variations in patch size.

Might the effect of patch size in biasing density judgments reflect an intrusion of number bias? Larger patches of the same density would, after all, contain more dots than smaller patches. It is certainly possible that decision biases are affected by multiple magnitude dimensions. However, given that density and size are covariant with viewing distance in normal circumstances, it seems likely that interactions between these two dimensions in middle or later stages of perceptual processing could be related by constancy mechanisms. Such constancy mechanisms are irrelevant for numerosity (which is invariant with viewing distance), except insofar as it may combine density and area information.

Experiment 2: Replication with a large sample of naïve observers

Experiment 1 demonstrated that density discrimination is more precise than numerosity discrimination using expert participants and stimuli with well-defined density. Given the novelty of the observation that density discrimination was superior to that of numerosity, it seemed best to replicate the experiment with a large group of naive observers in a pre-registered experiment (Simmons, Nelson, & Simonsohn, 2011). We recruited students completing a research requirement in an introductory psychology class, and adapted the paradigm so that a single session could measure both density and number.

Method

Open science

The experimental procedure, exclusion criteria, and analysis plan were preregistered on aspredicted.org: https://aspredicted.org/fv2y-28sf.pdf. All raw data files and a data key file are available on the OSF accessible at https://doi.org/10.17605/OSF.IO/7Z6XH.

Participants

Forty-eight Swarthmore College students participated for course credit (aged 18–22 years). Because of anticipated variability in the motivation of the participant pool, the registration called for retaining the 32 participants with the lowest overall geometric mean Weber fractions across all tasks. Participants wore corrective lenses as needed.

Apparatus

Three separate noise-shielded rooms were used. Each room housed an iMac with a resolution of 100 pixels/deg at the viewing distance of 66.5 cm. Chin and forehead rests were used to maintain the appropriate viewing distance. Custom software, programmed in MATLAB and PsychToolbox (Kleiner et al., 2007), controlled the displays. On-screen instructions led the participants through the various procedures.

Design

Each participant performed discrimination tasks for numerosity and for density similar to those in Experiment 1 (internal standard, presented only at the outset; feedback on every trial). One-half the participants did the number discrimination tasks first, and one-half did the density tasks first. For each task, perceptual precision was measured at each of 5 values (in random order) of numerosity (or density) that encompassed a logarithmic range representing a factor of 16. For number, these values were 20, 40, 80, 160, and 320 dots. For density, they were 1, 2, 4, 8, and 16 dots/deg². The staircases for each discrimination were controlled as in Experiment 1, but consisted of 76 trials instead of 100 per measurement. Psychometric fits were again made in log number or density space using logistic regression.

Dot patch specifications

The dot patch specifications were, for the most part, like those in Experiment 1 with respect to visual angle, with the exception that the dot diameters remained 4 to 16 pixels, which now subtended 0.04° to 0.16°. Additionally, variations in patch diameter relative to the standard were standardized for number and density, representing a proportional increment or decrement by up to 2^0.25, thus producing dot patches that could vary in nominal area by a factor of 2 within each measurement block. Note that, as in Experiment 1, the median size of the numerosity patches was constant across different numerosities, whereas the median size of the density patches was inversely proportional to density.

Results

We found that density discrimination was more precise than numerosity discrimination in a large pool of non-expert observers. Mean Weber fractions for density and numerosity are shown in Figure 4. The main pre-registered analysis was an ANOVA on the log of the Weber fractions with task type (density or number) and mean density value of the standards (1–16 dots/deg²) as predictors. As expected, Weber fractions were significantly lower for density (M = 0.143) than for numerosity (M = 0.163), F(1, 31) = 12.1, p = 0.001, η² = 0.11. The effect size of the overall difference was 0.58. This result replicates the primary observation of Experiment 1. There was also a significant effect of density, F(1,31) = 7.4, p = 0.011, η² = 0.09, reflecting that performance grew somewhat worse at the highest density (i.e., displays with the smallest dots). There was no reliable interaction between task and density value, F(1, 31) < 1, although it appears that the precision of numerosity judgments was somewhat better for 20 dots than for the larger numbers.

Figure 4.

Discrimination and patch size results of Experiment 2. (Left) Mean Weber fractions for the discrimination of density and of numerosity are shown based on data from 32 naïve participants. (The number values also the x axis can also be interpreted as the numeric values of the numerosity standards). (Right) Effects of random variations in patch size on each dimension (percent change). Standards errors of the means across participants (computed in log space) are shown.

Patch size effects

Our planned analyses included investigating whether density again showed an effect of patch size by splitting the data at the median patch size of each measurement block. As in Experiment 1, a larger patch size produced a significant increase in perceived density, which was represented by a smaller PSE (by 10%) for the larger patches than for the smaller ones, F(1, 31) = 14.9, p = 0.0005, η² = 0.12. For numerosity judgments, however, splitting the data by patch size tended to reveal an oppositely signed effect on perceived numerosity (with large sizes increasing the PSE by a factor of 3% on average—indicating a trend toward reduction in judged numerosity), F(1, 31) = 4.0, p = 0.054, η² = 0.06. There were no effects from the trial-to-trial variations of luminance contrast of the dots on either judgment.

Discussion

The present data replicate the observation of Experiment 1, again demonstrating that density perception is more precise than numerosity perception. This result was evident despite the fact that, with respect to bias, density judgments seemed, again, to be affected by patch size in a way that was not corrected by the presence of feedback. Given that larger patch sizes are normally accompanied by higher estimates of visual number (Dakin et al., 2011; Kreuger, 1972), our finding here that number judgments were, if anything, affected oppositely than density judgments suggests that participants overcompensated (at higher numerosities) when using trial feedback to calibrate their number judgments with respect to size. They did not fully correct for patch size effects on perceived density.

Could the use of more uniform textures have disadvantaged numerosity judgments? This seems unlikely. As mentioned, Burgess and Barlow (1983) found that number discrimination in central vision was improved when using relatively uniform textures (randomized within a square grid) as compared with random textures. Their data concerning this (from Burgess & Barlow, Figures 5 and 7) is replotted in Figure 5 in terms of Weber fractions.

Figure 5.

Numerosity discrimination data from Burgess and Barlow (1983). These replotted data from Burgess and Barlow (1983) show that their Weber fractions were lower for numerosity comparisons using smoothly dense textures than for random textures. Their entire display subtended 2.2° at their main viewing distance. The dots themselves subtended 0.13° (the minimum resolution of their display).

Burgess and Barlow (1983), after showing that the effects were not due to retinal density, concluded that “We cannot explain why observer variance changes with dot number.” However, the fact that performance was improved by using low-variance densities relative to random densities is consistent with the idea that reduced dot density variance and reduced patch size variance (which occurred as numerosity increased) may each, separately, improve the precision of numerosity judgments.

Using a single-stimulus task, with a patch size that varied from trial to trial, we did not observe a downward slope in Weber fractions either for number or for density judgments. It would appear that the downward slope first discovered by Burgess and Barlow (1983) and later by Anobile et al. (2014) is dependent on using fixed-size notional standard and comparison patches.

Examining the area–variance hypothesis

If visual perception ultimately integrates early density information with patch size to subsequently estimate numerosity, numerosity judgments ought to show sensitivity to patch size. Such sensitivity of numerosity to patch size has been demonstrated in the past (Dakin et al. 2011; Krueger, 1972; Portley & Durgin, 2019). Sensitivity to patch size may also be implicated in the surprising pattern of lower discrimination thresholds for higher numerosities first reported by Burgess and Barlow (1983) and later rediscovered by Anobile et al. (2014). Durgin (2025) noted that the variance of the area inside the convex hull of a random dot patch decreases precipitously as numerosity increases (using dots of uniform size). To evaluate the idea that this reduction in area variance might contribute to the lowering Weber fractions for higher numerosity comparisons, we re-examined the publicly available raw data (https://osf.io/6ax5j/?view_only=723ceb0b44da47dba99e56db12db02a9) originally reported in Durgin (2025).

In that prior study, Durgin (2025) reported data from 32 participants who made numerosity discriminations for 10 different numbers, ranging logarithmically from 10 to 226 dots. Each discrimination trial included both a standard patch and a comparison patch presented simultaneously to the left and right of fixation. (One-half of the participants made the discriminations between dot patches in regions that were nominally 5° × 10° [50 deg² in area] and one-half made the discriminations between the dot patches [using smaller dots] in regions that were 5.0° × 2.5° [12.5 deg² in area] at the same eccentricity, resulting in a 4-fold increase in retinal density.) Although Durgin (2025) noted that the variance in the area inside the convex hull of the randomly scattered dot patches used in the experiment decreased with increases in numerosity in a way that might account for a reduction in decision variance, he did not analyze whether variations in patch size had biased participant judgments in his data. Here, we used two versions of the data-splitting approach to determine the effect of relative patch size in this existing numerosity discrimination dataset.

We first used the log of the ratio between the areas (within the convex hulls) of the standard and comparison patches on each trial to split the data for each participant/numerosity based on whether this ratio was greater or less than 1 (positive or negative in log space). We computed the difference between the PSEs for each subset of the data, as well as the average difference (in log space) between the two size distributions. Because a Weber fraction represents variance in only one direction from the mean, we computed one-half of the difference between these PSEs as the magnitude of shift owing to the size difference, and half the size difference in log space as the magnitude of the size difference in each direction. The left panel of Figure 6 shows the average (one-half) size differences (gray line), the average (one-half) magnitude of the effect of size on numerosity judgments (black circles), with the overall pattern of Weber fractions (blue circles) included as a reference. Note that beyond about 28 items the Weber fractions in the data of Durgin (1995) showed a downward slope of 0.25 in log space, similar to that shown in the data of Burgess and Barlow (1983); this same range is where the reduction in actual patch size variance (represented by the gray line in the left panel of Figure 6) seems likely to play a role. This analysis shows that randomly larger patches were judged more numerous for most numerosities. Across the 8 numerosities from 10 to 113, the effect sizes (d) for the shifts in PSE were 0.89, 0.98, 0.86, 0.56, 0.65, 0.71, 0.57, and 0.64, respectively.

Figure 6.

Patch-size effects in the data from Durgin (2025). These plots are based on the data of 32 naïve participants (Durgin, 2025). The blue circles in each plot represent mean Weber fractions for numerosity, showing increasing precision beyond 28 dots. In the left plot, the gray line represents the average (half) magnitude of the difference between the convex hulls of the standard patch and the comparison patch (the result of random dot positioning on a trial), as a function of standard numerosity. The black dots represent half the magnitude of the difference in PSEs between the two subsets of data (comparison patch smaller than vs. larger comparison patch). In the right plot, the split-half size difference in the standard patches alone is represented by the gray line, and the black points show the difference in the PSEs based on splitting the data on the size of the standard patches on each trial. Standards errors of the means across participants (computed in log space) are shown.

Because the convex hull of the comparison patch would tend to be correlated with patch numerosity in Durgin's (2025) study, we repeated this splitting analysis using only the convex hull of the standard patch. The standard patch did not vary in numerosity across trials within each measurement block, but its convex hull would have varied randomly. For example, for a standard of 10 dots the mean of the randomly larger areas (based on a median split) was 1.5 times larger than the mean of the smaller areas (again clarifying why density judgments concerning such stimuli would tend to have unexplained stimulus variance adding to decision noise in prior studies). The effect of this difference ought to account for approximately half the shift in PSEs of the first analysis (i.e., the half-magnitude that was plotted in the left panel of Figure 6). The right panel of Figure 6 shows the shift in perception based only on splitting the data by the size of the standard patch. These effects are about the same as the half-magnitude of the shift owing to splitting the data by the ratio between the standard and comparison patches. This shows that the effect of patch size on numerosity judgements is truly a response to actual patch size and not a confound of numerosity.

Anobile et al. (2014) attributed the relatively high and relatively constant Weber fractions for low numerosities to pure number perception, devoid of effects of density. However, we have shown that it is in this number range (which was supposedly independent of density) that judgments of these numbers in the near periphery were most strongly impacted by random variations in the (actually presented) size of the standards (as well as of the comparison patches) in a study where the notional size of each patch was fixed. The size biases in this range represent about half the magnitude of the Weber fraction. In other words, the imputed number region seems to be particularly susceptible to random effects of patch size, consistent with the assumption that even in this range, the visual system is integrating estimates of local density and area to estimate numerosity.

General discussion

The present study set out to measure the precision of discrimination of density and numerosity using textures with well-defined densities. Both with a small set of expert observers and a large sample of naïve participants, discrimination of density was more precise than that of numerosity. Our empirical results confirm that local density information may underlie numerosity perception: If it is true that local density information is used for computing number via integration with area, the higher Weber fractions for number perception would simply reflect that number perception additionally needs to take area into account. Both the estimation of area and the integration of density and area might not be error-free calculations.

Additionally, the experiments in this study show that the downward-sloping Weber fractions previously attributed to density discrimination (Anobile et al., 2014) do not show up in tasks involving actual density discrimination. Specifically, this dependency did not emerge with a) densities that are well-defined and that avoid interpretive problems owing to local subregions with substantially different densities or random (unaccounted for) variation in actual mean density; and b) patch sizes that are variable, to disentangle density and number judgments. Using a pre-registered design and analysis plan, we have found that density discrimination is more precise than number perception.

Although we have shown that density discrimination is more precise than number discrimination, we have also seen that, even for well-defined densities, explicit density perception is strongly impacted by patch size. Patch size effects on perceived density were first reported by Dakin et al. (2011) using randomly scattered elements. Here we have replicated this phenomenon with low-variance stimuli, showing that that the result is not specific to more random textures. Given that retinal size and density covary with changes in viewing distance, it is interesting, but not surprising, that consciously available perceptual representations of density may (unconsciously) take patch size into account.

In contrast, patch-size effects on numerosity perception are hard to account for based on constancy mechanisms given that number is invariant with distance. Whereas our new data did not show typical evidence of patch size effects, which may have been masked by feedback-induced compensation, examination of existing number discrimination data show strong patch size effects even for numerosities as low as 10 dots. This seems easy to interpret in terms of the cognitive relevance of patch size to the estimation of visual number suggested by human sensitivity to differences in density.

Additionally, we have shown that claims that the precision of density discrimination is poor at low densities were based on analyses that ignored large random variation in actually presented densities. The interpretation of density discrimination in low-numerosity textures that do not take empirical area (and thus actual density) into account have been used to argue for a range of numerosities where numerosity discrimination is superior to density discrimination. These results now appear likely to be artifacts of the stimulus choices made by experimenters who did not take account of random (uncontrolled) variations in patch area when defining display density, but may also represent effect of sample-size in low number displays. Our results show that when participants are instructed to attend to trial-to-trial variations in density, they perform with greater precision than when they attend to trial-to-trial variations in numerosity (in stimuli where the two are decoupled by variations in display areas). Neither numerosity discrimination nor density discrimination show downward-sloping performance when display area is varied trial to trial. Given that the spatial statistics of the stimuli appear more critical than previously considered, further generalizations with additional stimulus types and underlying generation algorithms is strongly motivated.

None of our present findings contradict the possibility that judgments of numerosity may often make use of alternative, specialized segmentation processes or cognitive strategies. For example, there is compelling evidence that participants are able to evaluate approximate numerosities of segmented objects more easily than of their component parts (Franconeri, Bemis, & Alvarez, 2009). Observers are also able to simultaneously select (at least some) colored subsets within mixtures of dots (Becker, Dellinger, & Durgin, 2025; Cordes, Goldstein, & Heller, 2014; Halberda, Sires, & Feigenson, 2006; Lei & Reeves, 2018). Some forms of perceptual grouping aid estimation and comparison (e.g., Atkinson et al., 1976; Starkey & McCandliss, 2014). However, there is also evidence that, in the absence of count words (and potentially of mathematical training), the comparison of spatial numbers representing numerosities below 10 is notably poor (Gordon, 2004). This finding suggests that specialized strategies may develop through cultural transmission of technologies like count words (Frank, Everett, Fedorenko, & Gibson, 2008) and the learning of mappings of number to linear extents.

Density discrimination does not follow a square root function

One reviewer has asked us to explain a fairly large number of his papers that claim to show clear differences between “density” and “number.” We offer a brief discussion here. Many papers have been published that cite Anobile et al. (2014) as establishing that high numbers are processed by density mechanisms, but that low numbers are not. The argument was based on the idea that the discrimination of higher numbers follows a square root function attributable to density. However, as we have shown here, by isolating density from size and numerosity during testing, density discrimination does not intrinsically follow a square root function. This means that simply comparing performance for high numerosities and low numerosities is not an effective means of distinguishing density and number. Skepticism should be applied to any paper that made or makes arguments about density vs. numerosity based on solely on element number.

We have already offered some alternative explanations of the downward sloping Weber fractions observed by Anobile et al. (2014) and, earlier, by Burgess and Barlow (1983). We have supported one of these alternatives (i.e., owing to changes in patch size variability) by using archival data to show that, even for numerosities as low as 10 dots, variations in patch size influence numerosity judgments. Thus, any study that interprets differences in performance for high numerosities and low numerosities as representing differences between pure numerosity and density can be regarded as being based on a misconception broadly promulgated in the literature. This is not to question that differences in processing may exist at different ranges of numerosity, and for different types of stimuli (Durgin & Portley, 2023). But if a study makes a claim about density vs. numerosity by using low and high numerosities, it is not safe to assume that the difference observed categorically differentiate numerosity and density processing. Neither perceptual magnitude is limited to any particular range of number (beyond the subitizing range, say).

In general, our concern is that alternative accounts are rarely fully tested in making assertions about numerosity per se. As an example, Zimmermann (2018) used a large numerosity vs. small numerosity argument in a paper that did not clearly distinguish among alternative magnitude representations (e.g., did not test for effects of perceived size, for example). His paper, although carefully argued, also assumed that the density vs. number distinction proposed by Anobile et al. (2014) was appropriate to motivate the idea that number, per se, was adapted at low numbers. The aftereffects he measured indicated the involvement of larger receptive fields for numerosity 12 than numerosity 100. However, a numerosity of 12 does not single out pure number just by virtue of being small. Indeed, the broader receptive fields he measured for aftereffects at numerosity 12 seem potentially consistent with aftereffects on patch size analyzers, for example, and thus with magnitude representations. Owing to random size variation discussed elsewhere in this article, relative adapter patch size and patch numerosity in his low-numerosity adapters (he paired an adapting patch of 6 dots on one side with an adapting patch of 24 on the other), would have been confounded as an incidental effect of using smaller numerosities with random positioning.

Although we agree with the idea that numerosity judgments may often use other information besides density (including patch size; see also Aulet & Laurenco, 2021), and that density may be increasingly useful at higher numbers (Durgin, 1995), our sense of the literature is that those who argue for a “number sense” that is “direct” generally do not test for influences from or on other perceptual variables (except by a priori categorization, such as stipulating that low numbers are, by definition, sensed directly). This is clearly a problem, because when multiple perceptual variables are measured in adaptation experiments (Durgin & Martinez, 2024; Zimmerman & Fink, 2016) density and/or size are usually implicated in the perception of numerosity, even for fairly low numerosities, like 13 (Zimmerman & Fink, 2016) and 30 (Durgin et al., 2025). Moreover, there are patch-size effects for numerosity 10 in the paradigm used by Anobile et al. (2014). How numerosity is processed (e.g., directly or indirectly) is different from the question of whether numerosity is ultimately processed (e.g., DeWind, Adams, Platt, & Brannon, 2015).

Is there a special unmediated number sense?

Studies of the neural basis of number perception indicate that relevant information is present in the activity of early visual areas—typically in the form of local contrast energy (Fornaciai, Brannon, Woldorff, & Park, 2017; Harvey, 2025; Harvey & Dumoulin, 2017; Nieder, 2025; Park et al., 2016; Paul, van Ackooi, Ten Cate, & Harvey, 2022). Additional processing through the hierarchy of “early” to “mid” level visual stages likely extracts information that is closer to an explicit representation of density and/or numerosity (Fornaciai & Park, 2018). These studies do make it clear that a variety of relevant signals are available for use in estimating density and/or numerosity, but also highlight the fact that the stage of representation that is “read out” for judgments is critical, but not currently constrained by either physiological or psychophysical data. One important detail is that bidirectional density aftereffects (Durgin & Martinez, 2024; Sun et al., 2017) likely require an early divisive normalization stage (Park & Huber, 2022; see also Durgin, 2001).

Although many physiological studies have demonstrated rather explicit representations of number in associative, cognitive, and premotor parts of the parietal and frontal lobes, these studies shed light on the nature of the representation after read-out (or “decoding”) of the visual information (Cavdaroglu & Knops, 2019; Eger, Pinel, Dehaene, & Kleinschmidt, 2015; Harvey & Dumoulin, 2017; Harvey, Klein, Petridou, & Dumoulin, 2013; Nieder et al., 2002; Nieder & Miller, 2003; Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004; Tudusciuc & Nieder, 2007), and thus do not specify how numerosity is encoded in stages that would broadly be considered “sensory” in nature. A few human neuroimaging studies have investigated numerosity-dependent activation in human visual areas. Across these studies, some have tried to distinguish density from numerosity, some have manipulated various visual features but neglected to manipulate patch size, and most have focused on a rather restricted numerosity range when compared with the psychophysics. Although intriguing, this neuroimaging work does not yet offer strong leverage in resolving the debate regarding whether number estimates beyond the subitizing range are sensed directly, or are inferences based on density and area (DeWind, Adams, Platt, & Brannon, 2019; Park, DeWind, Woldorff, & Brannon, 2016; Paul et al., 2022; Zhang et al., 2025). Our hope is that the stimulus issues raised in this article will pave the way to more conclusive direct measurements of the sensory representations that lie at the heart of this debate.

Whereas behavioral studies of late readout of density and numerosity are challenging to interpret on their own, studies implicating early adaptation provide convergent evidence. In particular, there is growing evidence that strong numerosity aftereffects for reasonably large numbers (20 or more) may arise early in processing (i.e., are mainly retinotopic), and are consistent with adaptation occurring primarily at an early density representation (Durgin & Martinez, 2024; Durgin et al., 2025; Zimmermann, 2018). In contrast, for example, a recent report of object-centered number adaptation measured biases of about 10% (Myers, Firestone & Halberda, 2025), whereas classical density-based adaptation typically produces shifts by 50% in perceived numerosity (Durgin, 1995). Taken together, the experiments described here provide ample evidence for density as a visual building block of numerosity judgments, even at low numerosities (Zhang et al., 2025). They have also implicated the role of patch-size perception in numerosity judgments under conditions that have been staked out as being purely owing to a number sense (Anobile et al., 2014). Our analyses thus seem inconsistent with the idea of there being an unmediated “visual sense” of number.