It is often assumed that adaptation — a temporary change in sensitivity to a perceptual dimension following exposure to that dimension — is a litmus test for what is and is not a “primary visual attribute”. Thus, papers purporting to find evidence of number adaptation motivate a claim of great philosophical significance: That number is something that can be seen in much the way that canonical visual features, like color, contrast, size, and speed, can. Fifteen years after its reported discovery, (...) number adaptation’s existence seems to be nearly undisputed, with dozens of papers documenting support for the phenomenon. The aim of this paper is to offer a counterweight — to critically assess the evidence for and against number adaptation. After surveying the many reasons for thinking that number adaptation exists, we introduce several lesser-known reasons to be skeptical. We then advance an alternative account — the old news hypothesis — which can accommodate previously published findings while explaining various (otherwise unexplained) anomalies in the existing literature. Next, we describe the results of eight pre-registered experiments which pit our novel old news hypothesis against the received number adaptation hypothesis. Collectively, the results of these experiments undermine the number adaptation hypothesis on several fronts, whilst consistently supporting the old news hypothesis. More broadly our work raises questions about the status of adaptation itself as a means of discerning what is and is not a visual attribute. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 (...) or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

Research on human infants has begun to shed light on early-developing processes for segmenting perceptual arrays into objects. Infants appear to perceive objects by analyzing three-dimensional surface arrangements and motions. Their perception does not accord with a general tendency to maximize figural goodness or to attend to nonaccidental geometric relations in visual arrays. Object perception does accord with principles governing the motions of material bodies: Infants divide perceptual arrays into units that move as connected wholes, that move separately from one (...) another, that tend to maintain their size and shape over motion, and that tend to act upon each other only on contact. These findings suggest that o general representation of object unity and boundaries is interposed between representations of surfaces and representations of objects of familiar kinds. The processes that construct this representation may be related to processes of physical reasoning. (shrink)

Jerry Fodor deemed informational encapsulation ‘the essence’ of a system’s modularity and argued that human perceptual processing comprises modular systems, thus construed. Nowadays, his conclusion is widely challenged. Often, this is because experimental work is seen to somehow demonstrate the cognitive penetrability of perceptual processing, where this is assumed to conflict with the informational encapsulation of perceptual systems. Here, I deny the conflict, proposing that cognitive penetration need not have any straightforward bearing on the conjecture that perceptual processing is composed (...) of nothing but informationally encapsulated modules, the conjecture that each and every perceptual computation is performed by an informationally encapsulated module, and the consequences perceptual encapsulation was traditionally expected to have for a perception-cognition border, the epistemology of perception and cognitive science. With these points in view, I propose that particularly plausible cases of cognitive penetration would actually seem to evince the encapsulation of perceptual systems rather than refute/problematize this conjecture. (shrink)

Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...) along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

One of the most important abilities we have as humans is the ability to think about number. In this chapter, we examine the question of whether there is an essential connection between language and number. We provide a careful examination of two prominent theories according to which concepts of the positive integers are dependent on language. The first of these claims that language creates the positive integers on the basis of an innate capacity to represent real numbers. The second claims (...) that language’s function is to integrate contents from modules that humans share with other animals. We argue that neither model is successful. (shrink)

Many authors have posited an “object file” system, which underlies perceptual selection and tracking of objects. Several have proposed that this system internalizes principles specifying what counts as an object and relies on them during tracking. Here I consider a popular view on which the object file system is tuned to entities that satisfy principles of three-dimensionality, cohesion, and boundedness. I argue that the evidence gathered in support of this view is consistent with a more permissive view on which object (...) files select and track according to well-known perceptual organization criteria. Further evidence supplies positive support for the permissive view. (shrink)

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic (...) and non-symbolic approximate division. Subjects were presented with non-symbolic or symbolic dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children and adults were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction. (shrink)

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities, but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

Clarke and Beck argue that the approximate number system represents rational numbers, like 1/3 or 3.5. I think this claim is not supported by the evidence. Rather, I argue, ANS should be interpreted as representing natural numbers and ratios among them; and we should view the contents of these representations are genuinely approximate.

There are three independent properties of a mode of presentation of a number: being specific; being recognitional; and being canonical. A perceptual m.p. of the form that many Fs is specific although it is neither recognitional nor canonical. The literature has not distinguished noncanonical from nonspecific m.p.s of numbers. Ratios are fundamentally ratios of magnitudes.

In contrast to Clarke and Beck's claim that that the approximate number system represents rational numbers, we argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.

Clarke and Beck suggest that the ratio processing system may be a component of the approximate number system, which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

Clarke and Beck propose that the approximate number system (ANS) represents rational numbers. The evidence cited supports only the view that it represents ratios (and positive integers). Rational numbers are extensive magnitudes (i.e., sizes), whereas ratios are intensities. It is also argued that WHAT a system represents and HOW it does so are not as independent of one another as the authors assume.

Clarke and Beck use behavioural evidence to argue that approximate ratio computations are sufficient for claiming that the approximate number system represents the rationals, and the ANS does not represent the reals. We argue that pure behaviour is a poor litmus test for this problem, and that we should trust the psychophysical models that place ANS representations within the reals.

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

According to the dominant view in the literature, several numerical cognition phenomena are explained coherently and parsimoniously by the Approximate Number System model, which supposes the existence of an evolutionarily old, simple representation behind many numerical tasks. We offer an alternative account that proposes that only nonsymbolic numbers are processed by the ANS, while symbolic numbers, which are more essential to human mathematical capabilities, are processed by the Discrete Semantic System. In the DSS, symbolic numbers are stored in a network (...) of nodes, similar to conceptual or linguistic networks. The benefit of the DSS model and the benefit of the more general hybrid ANS–DSS framework are demonstrated using the crucial example of the distance and size effects of comparison tasks. (shrink)