Clarke and Beck (C&B) argue convincingly that the approximate number system (ANS) plays a vital role in allowing us to directly perceive number rather than numerosity, and we concur (Jones, Reference Jones2016, Reference Jones and Bangu2018). However, by adopting a representationalist stance and appealing to the notion of “modes of presentation,” they unnecessarily incur a heavy theoretical cost. By eschewing representationalism and the idea that the function of the ANS is to represent number, and instead adopting a radical enactivist stance (Hutto & Myin, Reference Hutto and Myin2012, Reference Hutto and Myin2017; Zahidi, Reference Zahidi2021; Zahidi & Myin, Reference Zahidi, Myin, Etzelmüller and Tewes2016), one can explain our direct perception of number with less philosophical baggage.

The issue of whether we have perceptual access to numerical properties is not new, and, over the last century, was largely seen by philosophers of mathematics to have been settled in the negative because of Frege's (Reference Frege1950, pp. 9–14, 27–32) infamous critique of Mill's empiricism, whereby the number we assign to a collection depends on our conceptualizations. As such, there is seemingly no room for number to be a mind-independent property that we directly perceive. This consensus has, more recently, begun to shift, in light of evidence that we have a natural capacity for directly apprehending numerical properties that is perceptual in nature (see Anobile, Cicchini, & Burr, Reference Anobile, Cicchini and Burr2016 for a review; Jones, Reference Jones and Bangu2018, pp. 150–152).

C&B convincingly argue that we perceive number directly and that the imprecision of the ANS is no reason to suggest that we perceive something other than numerical properties, for example, numerosities. However, this still leaves open the question of what we're perceiving when we perceive number. C&B's answer is that we perceive natural numbers as properties of collections (p. 10) and rational numbers as properties of ratios, yet this is unsatisfyingly trivial, because the terms “collection” and “ratio” merely refer to that to which (natural and rational) numerical properties can apply. Thus, their solution simply raises the question of how we perceive collections or ratios.

C&B take the idea that the ANS represents numbers to be the best explanation of the available evidence, but they neglect alternative non-representationalist explanations that accept our sensitivity to numerical properties without committing to any neural system representing those properties using particular modes of presentation. For example, it is possible to understand perception of numerical properties of collections as perception of affordances for engaging in various activities (Gibson, Reference Gibson1979; Jones, Reference Jones and Bangu2018; Kitcher, Reference Kitcher1984, pp. 11–12, 108). To “perceive a collection of apples as being seven in number” (p. 10) is to be sensitive to structural properties that are significant for a range of actions. The “seven-ness” is not a property of the apples, nor of the perceiver, but of what the perceiver can do with them. Rather than the ANS functioning to “keep count of whole items” (p. 34, emphasis removed), it plays a role in enabling actions such as counting. This approach is more closely aligned with recent evidence suggesting that the ANS is a sensorimotor system, rather than a simple number detector, because it is implicated in both numerical perception and numerical action, as well as interaction between the two (see Anobile, Arrighi, Castaldi, & Burr, Reference Anobile, Arrighi, Castaldi and Burr2021 for a review). This suggests that “the neuronal populations in the theory do not serve as representations of quantity, but serve as causal mediators between input and behavior” (Zahidi, Reference Zahidi2021, p. S537).

In making their case that the ANS represents, C&B rely heavily on a familiar philosophical conceit – the idea that represented items appear in specific guises or “modes of presentation.” This assumption puts them in position to explain how the ANS can be imprecise despite representing specific numbers. The notion of a mode of presentation originates in Fregean philosophy, where it is used to account for the sense, as opposed to the reference, of linguistic expressions. Several philosophers of mind make free and easy appeal to the idea that mental representations, and not just linguistically expressible thoughts, have modes of presentation. Even so, the distinction between the “sense” and “reference” of neural representations is an ad hoc construction without any independent justification. C&B try to motivate the use of modes of presentation by speaking of how the gustatory system might be thought to represent levels of sodium chloride (referent) via a “salty” mode of presentation (sense). However, this comparison is confusing, because the saltiness of sodium chloride is something experienced by an organism. There seems to be no obvious reason to suppose that there is a specific way that sodium chloride is presented to our sub-personal gustatory systems. By the same token, it is unclear what warrants assuming that a sub-personal neural system, such as the ANS, operates with a “mode of presentation,” or how we would be in a position to know which particular “mode of presentation” such a system would employ if it did. Positing “modes of presentation” does a lot of heavy lifting for C&B, but their appeal to that technical notion seems to be a “just so” solution, motivated by philosophical need rather than justified by independent empirical considerations.

There may be reasonable grounds for distinguishing the different ways organisms experience worldly targets or the ways people variously represent the same extension. What is not clear is that C&B can innocently assume that modes of presentation operate at the neural level. Nor is it clear how they justify attributing the particular modes of presentation to the ANS that they do. After all, when presented with supposedly imprecisely represented collections, we do not experience them as imprecisely presented to us. Instead, it is simpler to assume that we are sensitive to numerical properties, just not optimally so (as one would expect given physiological constraints).

The problems with C&B's representationalism stem from their assumption that the ANS's sole function is numerical perception. In essence, they assume that the ANS is some form of number module. However, the evidence suggests that the neural system that houses the ANS is involved in a whole host of other capacities, including motion processing, mental imagery, working memory, and the control of visuo-spatial attention and pointing and grasping motions (Culham & Kanwisher, Reference Culham and Kanwisher2001; Gillebert et al., Reference Gillebert, Mantini, Thijs, Sunaert, Dupont and Vandenberghe2011; Grefkes & Fink, Reference Grefkes and Fink2005; Simon, Mangin, Cohen, Le Bihan, & Dehaene, Reference Simon, Mangin, Cohen, Le Bihan and Dehaene2002), in line with the predictions of Anderson's neural reuse theory (Anderson, Reference Anderson2014; Hutto, Reference Hutto2019; Jones, Reference Jones2020; Penner-Wilger & Anderson, Reference Penner-Wilger and Anderson2013). Once one gives up on the idea that the ANS is a system solely for dealing with number, the idea that its job is to represent number is far less tempting.