In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have (...) so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

In this paper, I argue for three claims. The first is that the difference between analog and digital representation lies in the format and not the medium of representation. The second is that whether a given system is analog or digital will sometimes depend on facts about the user of that system. The third is that the first two claims are implicit in Haugeland's (1998) account of the distinction.

Recent evidence that young children seem to both understand false belief in one sense, but not in another, has led to two-systems theorizing about mindreading. By analyzing the most detailed two-systems approach in studying social cognition—the theory of mindreading defended by Ian Apperly and Stephen Butterfill—I argue that that even when dutifully constructed, two-systems approaches in social cognition struggle to adequately define the mindreading systems in terms of signature processing limits, an issue that becomes most apparent when investigating mindreading in (...) infancy. I end the article by developing several challenges that face any two-systems account of mindreading. (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)

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.

The notion of domain specificity plays a central role in some of the most important debates in cognitive science. Yet, despite the widespread reliance on domain specificity in recent theorizing in cognitive science, this notion remains elusive. Critics have claimed that the notion of domain specificity can't bear the theoretical weight that has been put on it and that it should be abandoned. Even its most steadfast proponents have highlighted puzzles and tensions that arise once one tries to go beyond (...) an initial intuitive sketch of what domain specificity involves. In this paper, we address these concerns head on by developing an account of what it means for a cognitive mechanism to be domain specific that overcomes the obstacles that have made domain specificity seem so problematic. We then apply this understanding of domain specificity to one of the key debates that it has figured prominently in—the rationalism-empiricism debate concerning the origins of cognitive traits—and introduce several related theoretical notions that work alongside domain specificity in helping to clarify what makes a view more (or less) rationalist. This example illustrates how the notion of domain specificity can, and should, continue to play a central role in ongoing debates in cognitive science. (shrink)

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A.. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with a representational system that clearly (...) does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles. (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)

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I (...) argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality. (shrink)

Provided we agree about the thing, it is needless to dispute about the terms. —David Hume, A treatise of human nature, Book 1, section VIIMap-like representations are frequently invoked as an alternative type of representational vehicle to a language of thought. This view presupposes that map-systems and languages form legitimate natural kinds of cognitive representational systems. I argue that they do not, because the collections of features that might be taken as characteristic of maps or languages do not themselves provide (...) scientifically useful information above and beyond what the individual features provide. To bring this out, I sketch several allegedly distinctive features of maps, and show how they can easily be grafted onto a simple logical language, resulting in a hybrid “manguage.” The ease with which these linguistic and map-like properties can be integrated into a single representational system raises the question of what work broader categories like language and map are doing. I maintain.. (shrink)

In this article we review the discussion over the thesis that language serves as an integrator of contents coming from different cognitive modules. After presenting the theoretical considerations, we examine two strands of empirical research that tested the hypothesis — spatial cognition and mathematical cognition. The idea shared by both of them is that each is composed of two separate modules processing information of a specific kind. For spatial thinking these are geometric information about the location of the object and (...) the information about the object’s properties such as color or size. For mathematical thinking, they are the absolute representation of small numbers and the approximate representation of numerosities. Language is said to integrate the two kinds of information within each of these domains, which the reviewed data demonstrates. In the final part of the paper, we offer some comments on the theoretical side of the discussion. (shrink)

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of (...) contrasting alternatives, children assign them a meaning distinct from some. In the “Don’t Give-a-Number task”, children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast – plus knowledge of a number’s membership in a count list – enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children’s interpretation of unknown numbers is further constrained by understanding numerical entailment relations – that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number inbetween – who either has or doesn’t have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words. (shrink)

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.

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)

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)

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 chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having noisy (...) or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (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)