Re-establishing the distinction between numerosity, numerousness, and number in numerical cognition (original) (raw)
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Non-numerical methods of assessing numerosity and the existence of the number sense
Journal of Numerical Cognition
In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existe...
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We agree that the ANS truly represents number. We endorse the authors’ conclusions on the arguments from confounds, congruency, and imprecision, though we disagree with many claims along the way. Here we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.
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Proceedings of the Royal Society B: Biological Sciences, 2014
Much evidence has accumulated to suggest that many animals, including young human infants, possess an abstract sense of approximate quantity, a number sense . Most research has concentrated on apparent numerosity of spatial arrays of dots or other objects, but a truly abstract sense of number should be capable of encoding the numerosity of any set of discrete elements, however displayed and in whatever sensory modality. Here, we use the psychophysical technique of adaptation to study the sense of number for serially presented items. We show that numerosity of both auditory and visual sequences is greatly affected by prior adaptation to slow or rapid sequences of events. The adaptation to visual stimuli was spatially selective (in external, not retinal coordinates), pointing to a sensory rather than cognitive process. However, adaptation generalized across modalities, from auditory to visual and vice versa. Adaptation also generalized across formats : adapting to sequential streams o...
The Number Sense Theory Needs More Empirical Evidence
Mind and Language, 2001
Educated adults are able to tell which of two numbers is the larger, add or multiply single-digit numbers, use numbers to estimate distances, indicate the time, exchange money, and so on. After a long and sophisticated schooling, some are able to solve complex arithmetical problems, and a few can even elaborate new theories of arithmetic. The central questions raised by Dehaene in his 'Number Sense Theory' (hereafter, NST) is 'where' this particular and sometimes exceptional ability to process the world in an arithmetical way comes from, where the corresponding functional architecture is located in the brain, and how it functions. Dehaene's main proposal is that, like many other species, humans are born with a domain specific and biologically-rooted ability to categorise things or events in their environment according to their numerosity. Such a general thesis is plausible and appealing, but, when examined in detail, Dehaene's proposition is to be considered as no more-and also, of course, no less-than a set of theoretical propositions with an excellent internal coherence but, in the present state of knowledge, lacking sufficiently strong empirical evidence on many points. The NST must also be seriously confronted with alternative proposals which, although generally less ambitious, can also account for many of the existing empirical data. Before discussing some aspects of the NST, we wish however to underline its two principal virtues. Firstly, the NST connects cognitive processing theories about numbers with brain functions. Or, in other words, it tries to localise the main components of the cognitive architecture in the brain as well as the flow of information between them (Dehaene and Cohen, 1995). The
Cortex
From our very early school years we start to realize that numbers govern much of our life. A glance at the headlines will tell us a crucial parliamentary bill was defeated by 149 votes, that inflation is steady at .9%, that the GNP has declined by 1% and so on. A flick of our telephone gives us the time (in digits) and date, the telephone numbers of our friends, with apps to furnish our bank balance, and how many steps we have made today. However, these symbolic representations of quantity, usually by Arabic numerals, capture only a small fragment of our daily experience with numerable quantities, and how these quantities guide our behaviour, and the ways we exploit our inner ability to "sense" the numerosity of these quantities. By showing that birds can perform both simultaneous visuo-spatial and temporal-sequential coding of the numerosity of simple visual items (clouds of dots), the German zoologist Otto Koehler (1941; 1950) was among the first to suggest that the symbolic mathematical competence that characterises much human activity might be grounded in phylogenetically older systems that allow approximate, but behaviourally adaptive, estimates of numerosity. During biological evolution these rudimentary mathematical abilities might have been crucial for survival and adaptation by allowing, for example, the recognition and memorization of environments with more or fewer food items, or by favouring
Numbers as Cognitive Tools (PhD dissertation)
Vrije Universiteit Amsterdam, 2021
Do numbers exist? Most of the answers to this question presented in the literature of the last decades have relied on a priori methods of investigation, where scientific data and theories about the human experience of numbers are irrelevant. These a priori approaches, however, have been inconclusive. In this dissertation, I adopt an empirically informed approach in which scientific descriptions of the human experience of numbers—as provided by cognitive sciences, linguistics, developmental psychology, and mathematics education—provide valuable information on the existence and status of what we call “numbers.” These scientific descriptions allow for the conclusion that numbers, conceived of as independent, non-spatiotemporal objects, do not exist. What exist are certain human-made techniques which engender in us the idea that a special class of objects called numbers exists. I show that, just as counting procedures and other arithmetical operations are cognitive tools that allow us to go beyond the limits of our genetically endowed cognitive skills, the very idea that numbers exist as independent objects is a cognitive tool that facilitates calculation—in other words, a useful reification. The ontological hypothesis suggested by the scientific description of the human experience of numbers is that operations such as counting and calculating procedures are the objective subject matter underlying arithmetic, rather than a putative class of non-spatiotemporal objects. Thus, the claim is that applied and pure arithmetical statements are true of the counting procedure and other arithmetical operations, rather than true of non-spatiotemporal numbers. In contrast to other attempted answers to the question of the ontological status of numbers, the hypothesis defended in this dissertation is accountable towards empirical data, and can thus be improved or refuted on an empirical basis.
A cognitive taxonomy of numeration systems
2007
Abstract In this paper, we study the representational properties of numeration systems. We argue that numeration systems are distributed representations—representations that are distributed across the internal mind and the external environment. We analyze number representations at four levels: dimensionality, dimensional representations, bases, and symbol representations.