Forget number for a moment. Consider another question: Do we perceive objects? The answer to this question must be an unambiguous “yes!”; our understanding of attention depends on the notion of “objects.”

But what is an object? Let's start with a rectangle. That's obviously an object. If we cut that rectangle right down the middle to form two separate squares, now we have two objects. Simple enough. What about cases in between? Suppose we connect our two separated squares with a thin line. Is this one object or two? If your answer is “two,” then ask: How thick does that line have to become before your answer becomes “one”? Now, consider the opposite. Start with a single rectangle, and cut a hole out of the middle of it (and imagine that this hole is as wide as the gap between the two squares you imagined before). Is that still one object? If so, how tall does that hole have to become before the rectangle becomes two objects? These examples hardly scratch the surface of objecthood edge cases.

Both the perception of objecthood and the perception of number blur the lines between the continuous and the discrete. On the one hand, the essence of objects is that they discretize attention; on the other hand, multiple independent cues simultaneously influence our impressions of objecthood (e.g., Feldman, Reference Feldman2007), resulting in continuous effects on attention and on spatial/temporal perception (e.g., Yousif & Scholl, Reference Yousif and Scholl2019). Similarly, it seems as if number must be discrete (what would it mean to perceive 16.5 objects?); yet, at the same time, we often talk about “numerosity” as if to imply that we perceive something vaguer and more imprecise than a discrete number. The perception of number and the perception of objecthood share this same ambiguity, yet, for some reason, there aren't multiple BBS papers (see also Leibovich, Katzin, Harel, & Henik, Reference Leibovich, Katzin, Harel and Henik2017) discussing whether we perceive objects – and nobody has yet felt the need to coin the term “objectosity.” What's the difference?

Here's my best guess: This boils down to the fact that objecthood has consequences; objecthood influences attention, and that's measurable (e.g., Egly, Driver, & Rafal, Reference Egly, Driver and Rafal1994). If we want to know whether objecthood was manipulated, we can simply ask whether object-based attention effects are attenuated. Imagine though if we did not have measures of object-based attention. How would we determine that objecthood is continuous – having observers make quick key press judgments about which of two things was more “object-like”? We would never be satisfied with such evidence. We may argue endlessly about this confound, or that one. Every few years, someone would come up with a new confound to argue over. We'd devote several BBS papers to discuss this deep, crucial matter of whether the visual system actually perceives objects. So it is with number.

How should this influence how we think about number in relation to other spatial properties? Traditionally, we think of number and area as things that ought to be perceived separately. There is a general thought that if area interferes with number perception, we must not be perceiving number directly, or veridically (see “The Argument from Confounds”; but see also Yousif & Keil, Reference Yousif and Keil2020). Clarke and Beck's argument indirectly raises a radical possibility: If the number system represents fractions, does that mean that the number system represents partial objects (i.e., can distinguish between 16 vs. 16.5 objects)?

If it is true that the number system represents partial objects, it would force us to reconsider how we think about confounds in quantity perception tasks. Consider again this idea that the visual system is tasked with counting objects. Now, imagine a display with 20 identical rectangles (see Fig. 1). Suppose we conduct a numerosity estimation task on this display, and we find that observers perceive and represent the display as having approximately 20 things. Now, imagine that we have a similar display, except that seven of the 20 rectangles have been cut in half, such that what remains in the display are 13 full rectangles and 7 half rectangles. Stated differently: there remain 16.5 full rectangles. We should expect that this latter display is perceived as less numerous, and we would traditionally explain this effect in terms of a congruency between number and area. But what if, instead, the number system is just representing the fact that there are partial entities in the display? What if the number system is representing the rational number 16.5?

Figure 1. Two sets, one with 20 “full” objects, and one with 16.5 “full” objects. Does the number system represent these partial objects?

If true, this leads to the novel prediction above: that area ought to influence number perception (insofar as area reflects partial wholes). After all, spatial cues influence the perception objecthood (Franconeri, Bemis, & Alvarez, Reference Franconeri, Bemis and Alvarez2009; Yu, Xiao, Bemis, & Franconeri, Reference Yu, Xiao, Bemis and Franconeri2019), and the visual system must be counting objects. This would be ecologically realistic. Seven half meals are equivalent to 3.5 full meals; it could be argued that this latter quantity, and not the former, is the better one to represent. (At the extremes, this would not be viable. One massive object probably ought not to be equated with 100 tiny objects of equivalent size. But, under most circumstances, similar objects are often of a fairly similar size. Edges cases such as these likely would not occur very often in the natural environment. Or, maybe they do! My suggestion is only that future research could consider this intriguing possibility.)

This possibility may or may not pan out empirically, yet it is one of many ways that Clarke and Beck's suggestion could radically alter how we think about quantity perception moving forward.