Human cognition is striking in its brilliance and its adaptability. How do we get that way? How do we move from the nearly helpless state of infants to the cognitive proficiency that characterizes adults? In this paper I argue, first, that analogical ability is the key factor in our prodigious capacity, and, second, that possession of a symbol system is crucial to the full expression of analogical ability.

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist (...) hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (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)

I argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system, whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.

What is the relationship between space and time in the human mind? Studies in adults show an asymmetric relationship between mental representations of these basic dimensions of experience: Representations of time depend on space more than representations of space depend on time. Here we investigated the relationship between space and time in the developing mind. Native Greek‐speaking children watched movies of two animals traveling along parallel paths for different distances or durations and judged the spatial and temporal aspects of these (...) events (e.g., Which animal went for a longer distance, or a longer time?). Results showed a reliable cross‐dimensional asymmetry. For the same stimuli, spatial information influenced temporal judgments more than temporal information influenced spatial judgments. This pattern was robust to variations in the age of the participants and the type of linguistic framing used to elicit responses. This finding demonstrates a continuity between space‐time representations in children and adults, and informs theories of analog magnitude representation. (shrink)

The dual-code proposal of number representation put forward by Cohen Kadosh & Walsh (CK&W) accounts for only a fraction of the many modes of numerical abstraction. Contrary to their proposal, robust data from human infants and nonhuman animals indicate that abstract numerical representations are psychologically primitive. Additionally, much of the behavioral and neural data cited to support CK&W's proposal is, in fact, neutral on the issue of numerical abstraction.

The lack of consensus on how to characterize humans’ capacity for belief reasoning has been brought into sharp focus by recent research. Children fail critical tests of belief reasoning before 3 to 4 years (Wellman, Cross, & Watson, 2001; Wimmer & Perner, 1983), yet infants apparently pass false belief tasks at 13 or 15 months (Onishi & Baillargeon, 2005; Surian, Caldi, & Sperber, 2007). Non-human animals also fail critical tests of belief reasoning but can show very complex social behaviour (e.g., (...) Call & Tomasello, 2005). Fluent social interaction in adult humans implies efficient processing of beliefs, yet direct tests suggest that belief reasoning is cognitively demanding, even for adults (e.g., Apperly, Samson & Humphreys, 2005). We interpret these findings by drawing an analogy with the domain of number cognition, where similarly contrasting results have been observed. We propose that the success of infants and non-human animals on some belief reasoning tasks may be best explained by a cognitively efficient but inflexible capacity for tracking belief-like states. In humans this capacity persists in parallel with later-developing, more flexible but more cognitively demanding theory of mind abilities. (shrink)

Cognitive partitions are useful. The notion of numerical identity helps us induce them. Consider, for instance, the role of identity in representing an equivalence relation like taking the same train. This expressive function of identity has been largely overlooked. Other possible functions of the concept have been over-emphasized. It is not clear that we use identity to represent individual objects or quantify over collections of them. Understanding what the concept is good for looks especially urgent in light of the fact (...) that numerical identities themselves are tangibly trifling. (shrink)

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure (...) magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements. (shrink)

Realists about animal cognition confront a puzzle. If animals have real, contentful cognitive states, why can’t anyone say precisely what the contents of those states are? I consider several possible resolutions to this puzzle that are open to realists, and argue that the best of these is likely to appeal to differences in the format of animal cognition and human language.

Empirical discussions of mental representation appeal to a wide variety of representational kinds. Some of these kinds, such as the sentential representations underlying language use and the pictorial representations of visual imagery, are thoroughly familiar to philosophers. Others have received almost no philosophical attention at all. Included in this latter category are analogue magnitude representations, which enable a wide range of organisms to primitively represent spatial, temporal, numerical, and related magnitudes. This article aims to introduce analogue magnitude representations to a (...) philosophical audience by rehearsing empirical evidence for their existence and analysing their format, their content, and the computations they support. 1 Background1.1 Evidence of analogue magnitude representations1.2 Weber’s law1.3 Scepticism about analogue magnitude representations2 Format2.1 Carey’s analogy2.2 Neural realization2.3 Analogue representation2.4 Analogue magnitude representation components3 Content3.1 Do analogue magnitude representations have representational content?3.2 What do analogue magnitude representations represent?3.3 What content types do analogue magnitude representations have?4 Computations4.1 Arithmetic computation4.2 Practical deliberation5 Conclusion. (shrink)

Strong nativist views about numerical concepts claim that human beings have at least some innate precise numerical representations. Weak nativist views claim only that humans, like other animals, possess an innate system for representing approximate numerical quantity. We present a new strong nativist model of the origins of numerical concepts and defend the strong nativist approach against recent cross-cultural studies that have been interpreted to show that precise numerical concepts are dependent on language and that they are restricted to speakers (...) of languages with the right kind of structure. (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)

Descartes argued that productivity, namely our ability to generate an unlimited number of new thoughts or ideas from previous ones, derives from a single undividable source in the human soul. Cognitive scientists, in contrast, have viewed productivity as a modular phenomenon. According to this latter view, syntactic, semantic, musical or visual productivity emerges each from their own generative engines in the human brain. Recent evidence has, however, led some authors to revitalize the Cartesian theory. According to this view, a single (...) source or a single mechanism in the human brain produces productivity in every cognitive domain, whether in the domain of music, semantics or syntax. In this article, we will address recent evidence concerning the single source hypothesis from brain-imaging studies, linguistics, cognitive theories of music perception, biology of cognition and cognitive development, along with several objections that have been presented against this hypothesis. We formulate two versions of the Cartesian theory which combine the more recent computational theory of cognition with Descartes' view on productivity. (shrink)

: Fodor argues that perceptual representations are a subset of iconic representations, which are distinguished from symbolic/discursive representations. Iconic representations are nonconceptual and they do not support the abilities afforded by concepts. Iconic representations, for example, cannot support object individuation. If someone thinks that perception or some of its parts has imagistic NCC, they face the following dilemma. Either they will have to accept that this NCC does not allow for object individuation, but it represents instead conglomerations of properties and (...) at some stage of visual processing it must interface with cognition and its conceptual capacities for the visual objects to be individuated. Or, they will have to hold that the imagistic, NCC of perception, allows for object individuation. I opt for the second thesis because I think there is strong empirical evidence that objects are individuated during early vision. I also think that early vision individuates objects by means of, what I had previously called nonconceptual perceptual demonstrative reference. I argue, first, why Fodor’s view that iconic NCC does not enable object individuation is false. I also argue, contra Fodor, that early vision allows the perception of the cardinality of sets of objects. Keywords : Early Vision; Analog Representations; Object Individuation; Arithmetic Cognition; Cardinality of Sets L'individuazione di oggetti mediante il contenuto iconico: come è rappresentata la numerosità nella rappresentazione iconica? Riassunto: Per Fodor le rappresentazioni percettive sono un sottoinsieme delle rappresentazioni iconiche, distinte dalle rappresentazioni simbolico/discorsive. Le rappresentazioni iconiche sono nonconcettuali e non supportano le abilità richieste dai concetti. Le rappresentazioni iconiche, per esempio, non supportano l'individuazione di oggetti. Se si pensa che la percezione o qualche sua parte abbia un contenuto nonconcettuale come immagine, si ci si imbatte nel seguente dilemma. O si accetta che il NCC non permetta di individuare oggetti, ma che rappresenti conglomerati di proprietà e che si interfacci con la cognizione e le sue capacità concettuali, per individuare gli oggetti visivi. O si dice che l'immagine, il NCC, della percezione consenta di individuare oggetti. Propendo per la seconda tesi, poiché penso che ci sia solida evidenza empirica per cui gli oggetti vengono individuati durante le prime fasi della visione, in cui l’individuazione avviene mediante quanto definito come riferimento dimostrativo del contenuto nonconcettuale percettivo. Chiarirò le ragioni per cui la concezione di Fodor, per cui il contenuto nonconcettuale iconico non supporta l’individuazione di oggetti, è falsa. Inoltre, contra Fodor, sosterrò che le prime fasi della visione permettono la percezione della cardinalità di insiemi di oggetti. Parole chiave: Prime fasi della visione; Rappresentazioni analogiche; Individuazione di oggetti; Cognizione aritmetica; Cardinalità degli insiemi. (shrink)

The view that special science properties are multiply realizable has been attacked in recent years by Shapiro, Bechtel and Mundale, Polger, and others. Focusing on psychological and neuroscientific properties, I argue that these attacks are unsuccessful. By drawing on interspecies physiological comparisons I show that diverse physical mechanisms can converge on common functional properties at multiple levels. This is illustrated with examples from the psychophysics and neuroscience of early vision. This convergence is compatible with the existence of general constraints on (...) the evolution of cognitive systems, and does not involve any ad hoc typing of coarse-grained higher level properties. The mechanisms that realize these common higher level properties are really distinct by the criteria laid down by critics of multiple readability. Finally, I present an account of how such functional properties might constitute special science kinds by playing a central explanatory role in a range of cognitive models. Behavioral science kinds in particular are the functionally defined constituents picked out by our most successful models of the multilevel systems and mechanisms that explain cognitive capacities. (shrink)

Recent years have witnessed an enormous increase in behavioral and neuroimaging studies of numerical cognition. Particular interest has been devoted toward unraveling properties of the representational medium on which numbers are thought to be represented. We have argued that a correct inference concerning these properties requires distinguishing between different input modalities and different decision/output structures. To back up this claim, we have trained computational models with either symbolic or nonsymbolic input and with different task requirements, and showed that this allowed (...) for an integration of the existing data in a consistent manner. In later studies, predictions from the models were derived and tested with behavioral and neuroimaging methods. Here we present an integrative review of this work. (shrink)

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)

Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical or proportional quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are fundamental in encoding the meaning (...) of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity biased the response direction. Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers. (shrink)