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.

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence that these (...) ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (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)

Research on the capacity to understand others' minds has tended to focus on representations ofbeliefs,which are widely taken to be among the most central and basic theory of mind representations. Representations ofknowledge, by contrast, have received comparatively little attention and have often been understood as depending on prior representations of belief. After all, how could one represent someone as knowing something if one does not even represent them as believing it? Drawing on a wide range of methods across cognitive science, (...) we ask whether belief or knowledge is the more basic kind of representation. The evidence indicates that nonhuman primates attribute knowledge but not belief, that knowledge representations arise earlier in human development than belief representations, that the capacity to represent knowledge may remain intact in patient populations even when belief representation is disrupted, that knowledge (but not belief) attributions are likely automatic, and that explicit knowledge attributions are made more quickly than equivalent belief attributions. Critically, the theory of mind representations uncovered by these various methods exhibits a set of signature features clearly indicative of knowledge: they are not modality-specific, they are factive, they are not just true belief, and they allow for representations of egocentric ignorance. We argue that these signature features elucidate the primary function of knowledge representation: facilitating learning from others about the external world. This suggests a new way of understanding theory of mind – one that is focused on understanding others' minds in relation to the actual world, rather than independent from it. (shrink)

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision – and show that none succeed. (...) We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position. (shrink)

The book is addressed to all readers interested in cognitive science, and especially in research combining a logical analysis with psychological, linguistic and neurobiological approaches. The publication is the result of a collaboration between the Department of Cognitive Science at University of Lodz and the Department of Cognitive Science at Lund University. It is intended to provide a comprehensive presentation of the key research issues undertaken in both Departments, including considerations on meaning, natural language and reasoning, linguistic as well as (...) numerical competence, the theory of decision making, modelling of conceptual representations, cognitive and game theoretic approach to social interactions. (shrink)

This paper refines a controversial proposal: that core systems belong to a perceptual kind, marked out by the format of its representational outputs. Following Susan Carey, this proposal has been understood in terms of core representations having an iconic format, like certain paradigmatically perceptual outputs. I argue that they don’t, but suggest that the proposal may be better formulated in terms of a broader analogue format type. Formulated in this way, the proposal accommodates the existence of genuine icons in perception, (...) and avoids otherwise troubling objections. (shrink)

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating an account of numbers as properties of collections, work on the philosophy of absences, and recent (...) work in numerical cognition and ontogenetic studies. (shrink)

People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often (...) conforms to abstract mathematical principles, it is typically implemented by perceptual and sensorimotor engagement with concrete environmental structures. (shrink)

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)

This paper will discuss the origin of the human mind, and the qualitative discontinuity between human and animal cognition. We locate the source of this discontinuity within the language faculty, and thus take the origin of the mind to depend on the origin of the language faculty. We will look at one such proposal put forward by Hauser et al. (Science 298:1569-1579, 2002), which takes the evolution of a Merge trait (recursion) to solely explain the differences between human and animal (...) cognition. We argue that the Merge-only hypothesis fails to account for various aspects of the human mind. Instead we propose that the process of lexicalisation is also unique to humans, and that this process is key to explaining the vast qualitative differences. We will argue that lexicalisation is a process through which concepts are reformatted to be able to take on semantic features and to take part in grammatical relations. These are both necessary conditions for a grammatical mind and the increased ability to express conceptual content. We therefore propose a possible explanans for the discontinuity between humans and animals, namely that merge with lexicalisation (and consequently semantic features and grammatical relations) is a minimal requirement for the human mind. (shrink)

Recent evidence from cognitive neuroscience suggests that certain cognitive processes employ perceptual representations. Inspired by this evidence, a few researchers have proposed that cognition is inherently perceptual. They have developed an innovative theoretical approach that rests on the notion of perceptual simulation and marshaled several general arguments supporting the centrality of perceptual representations to concepts. In this article, I identify a number of weaknesses in these arguments and defend a multiple semantic code approach that posits both perceptual and non-perceptual representations.

People’s awareness of tactile stimuli has been investigated in far less detail than their awareness of stimuli in other sensory modalities. In an attempt to fill this gap, we provide an overview of studies that are pertinent to the topic of tactile consciousness. We discuss the results of research that has investigated phenomena such as “change blindness”, phantom limb sensations, and numerosity judgments in tactile perception, together with the results obtained from the study of patients affected by deficits that can (...) adversely affect tactile perception such as neglect, extinction, and numbsense. The similarities as well as some of the important differences that have emerged when visual and tactile conscious information processing have been compared using similar experimental procedures are highlighted. We suggest that conscious information processing in the tactile modality cannot be separated completely from the more general processing of spatial information in the brain. Finally, the importance of considering tactile consciousness within the larger framework of multisensory information processing is also discussed. (shrink)

This chapter presents an overview of several recent developments in vision science, and outlines some of their implications for the management of visual attention in graphic displays. These include ways of sending attention to the right item at the right time, techniques to improve attentional efficiency, and possibilities for offloading some of the processing typically done by attention onto nonattentional mechanisms. In addition it is argued that such techniques not only allow more effective use to be made of visual attention, (...) but also open up new possibilities for human-machine interaction. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

This article critically examines the contemporary resurgence of empiricism (or “neo-empiricism”) in philosophy, psychology, neuropsychology, and artificial intelligence. This resurgence is an important and positive development. It is the first time that this centuries-old empiricist approach to cognition is precisely formulated in the context of cognitive science and neuroscience. Moreover, neo-empiricists have made several findings that challenge amodal theories of concepts and higher cognition. It is argued, however, that the theoretical foundations of and the empirical evidence for neo-empiricism are not (...) as strong as is usually claimed by its proponents. The empirical evidence for and against neo-empiricism is discussed in detail. (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)

Valence is a central component of all affective states, including pains, pleasures, emotions, moods, and feelings of desire or repulsion.This paper has two main goals. One is to suggest that enough is now known about the causes, consequences, and properties of valence to indicate that it forms a unitary natural-psychological kind, one that seemingly plays a fundamental role in motivating all kinds of intentional action. If this turns out to be true, then the correct characterization of the nature of valence (...) becomes an urgent philosophical issue. There appear to be just two accounts that have the required generality. According to one, valence is a nonconceptual representation of value. According to the other, valence is an intrinsic qualitative property of experience. The second goal of the paper is to contrast and evaluate these two views of the nature of valence, drawing on the relevant empirical findings. Overall, I suggest that the representational account is more plausible. (shrink)

In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and (...) neuropsychology) that suggests that the human capacity for mathematics is a category-specific domain of knowledge, hard-wired in the brain, which can be explained as the result of natural selection. (shrink)

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

A number of two-systems accounts have been proposed to explain the apparent discrepancy between infants’ early success in nonverbal mindreading tasks, on the one hand, and the failures of children younger than four to pass verbally-mediated false-belief tasks, on the other. Many of these accounts have not been empirically fruitful. This paper focuses, in contrast, on the two-systems proposal put forward by Ian Apperly and colleagues. This has issued in a number of new findings. The present paper shows that the (...) theoretical arguments offered in support of Apperly’s account are nevertheless unconvincing, and that the data can be explained in other terms. A better view is that there is just a single mindreading system that exists throughout, but which undergoes gradual conceptual enrichment through infancy and childhood. This system can be used in ways that do, or do not, draw on executive resources and/or working memory. (shrink)

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a (...) few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line. (shrink)

The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.

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.

The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...) 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

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)

This paper responds to Alva Noë’s general critique of Radical Enactivism. In particular, it responds to his claim that Radical Enactivism denies experience, presence and the world. We clarify Radical Enactivism’s actual arguments and positive commitments in this regard. Finally, we assess how Radical Enactvism stands up in comparison with Noë’s own version of Sensorimotor Knowledge Enactivism.

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that can deal (...) with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)

In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and (...) neuropsychology) that suggests that the human capacity for mathematics is a category-specific domain of knowledge, hard-wired in the brain, which can be explained as the result of natural selection. (shrink)

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)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

Perceptual Dogmatism holds that if it perceptually seems to S that p, then S has immediate prima facie justification for the belief that p. Various philosophers have made the notion of a perceptual seeming more precise by distinguishing perceptual seemings from both sensations and beliefs to accommodate a) the epistemic difference between perceptual judgments of novices and experts, and, b) the problem of the speckled hen. Using somewhat different terminology, perceptual seemings are supposed to be high-level percepts instead of low-level (...) sensations. I argue that although it is right that perceptual seemings should not be identified with sensations, they should also not be identified with percepts. There is no strong reason to assume that sensations and percepts exist as separate conscious states, and it appears therefore best to identify perceptual seemings simply with perceptual experiences interpreted as entities incorporating aspects from both sensations and percepts. However, even with this plausible identification in hand, the speckled hen will remain problematic for PD. (shrink)

I reply to comments by David Miguel Gray and Grant Gillett concerning my paper, ‘The Generality Constraint and the Structure of Thought’.