The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...) ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

Experimental studies indicate that nonhuman animals and infants represent numerosities above three or four approximately and that their mental number line is logarithmic rather than linear. In contrast, human children from most cultures gradually acquire the capacity to denote exact cardinal values. To explain this difference, I take an extended mind perspective, arguing that the distinctly human ability to use external representations as a complement for internal cognitive operations enables us to represent natural numbers. Reviewing neuroscientific, developmental, and anthropological evidence, (...) I argue that the use of external media that represent natural numbers (like number words, body parts, tokens or numerals) influences the functional architecture of the brain, which suggests a two-way traffic between the brain and cultural public representations. (shrink)

We present a model of how counting is learned based on the ability to perform a series of specific steps. The steps require conceptual knowledge of three components: numerosity as a property of collections; numerals; and one-to-one mappings between numerals and collections. We argue that establishing one-to-one mappings is the central feature of counting. In the literature, the so-called cardinality principle has been in focus when studying the development of counting. We submit that identifying the procedural ability to count with (...) the cardinality principle is not sufficient, but only one of the several steps in the counting process. Moreover, we suggest that some of these steps may be facilitated by the external organization of the counting situation. Using the methods of situated cognition, we analyze how the balance between external and internal representations will imply different loads on the working memory and attention of the counting individual. This analysis will show that even if the counter can competently use the cardinality principle, counting will vary in difficulty depending on the physical properties of the elements of collection and on their special arrangement. The upshot is that situated factors will influence counting performance. (shrink)

In a dynamic world, mechanisms allowing prediction of future situations can provide a selective advantage. We suggest that memory systems differ in the degree of flexibility they offer for anticipatory behavior and put forward a corresponding taxonomy of prospection. The adaptive advantage of any memory system can only lie in what it contributes for future survival. The most flexible is episodic memory, which we suggest is part of a more general faculty of mental time travel that allows us not only (...) to go back in time, but also to foresee, plan, and shape virtually any specific future event. We review comparative studies and find that, in spite of increased research in the area, there is as yet no convincing evidence for mental time travel in nonhuman animals. We submit that mental time travel is not an encapsulated cognitive system, but instead comprises several subsidiary mechanisms. A theater metaphor serves as an analogy for the kind of mechanisms required for effective mental time travel. We propose that future research should consider these mechanisms in addition to direct evidence of future-directed action. We maintain that the emergence of mental time travel in evolution was a crucial step towards our current success. (shrink)

Why Theories of Concepts Should Not Ignore the Problem of Acquisition.

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 considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (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)

Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a (...) dual-task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic. (shrink)

Recent research has suggested that the Pirahã, an Amazonian tribe with a number-less language, are able to match quantities > 3 if the matching task does not require recall or spatial transposition. This finding contravenes previous work among the Pirahã. In this study, we re-tested the Pirahãs’ performance in the crucial one-to-one matching task utilized in the two previous studies on their numerical cognition, as well as in control tasks requiring recall and mental transposition. We also conducted a novel quantity (...) recognition task. Speakers were unable to consistently match quantities > 3, even when no recall or transposition was involved. We provide a plausible motivation for the disparate results previously obtained among the Pirahã. Our findings are consistent with the suggestion that the exact recognition of quantities > 3 requires number terminology. (shrink)