Abstract
On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), 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(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g., 7), but also non-natural rational numbers (e.g., 3.5). It does not represent irrational numbers (e.g., √2), 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.
Highlights
On a orthodox view, humans and many non-human animals possess a number sense, or approximate number system (ANS), that affords a primitive and prelinguistic capacity to represent number
Number vs. Numerosity We have considered three arguments which purport to undermine the conjecture that the number sense represents number and found no reason to reject this conjecture
Earlier we argued that the ANS represents numbers, not non-numerical magnitudes, because it is sensitive to the second-order character of numbers, which is an essential property of numbers but not non-numerical magnitudes
Summary
On a orthodox view, humans and many non-human animals possess a number sense, or approximate number system (ANS), that affords a primitive and prelinguistic capacity to represent number. Since displays with small breaks and displays of items connected with thin lines differ only slightly with respect to total surface area, spatial frequency, and other non-numerical magnitudes, these studies are (again) indicative of a genuine sensitivity to the number of objects observed. If the fact that numerical judgments are influenced by area and density shows that number is not represented by an ANS, there should be equal or greater evidence that area and density are not represented either In this way, the argument from congruency leads to an implausible skepticism about the perceptual representation of magnitudes quite generally. Proponents of the argument from imprecision still require an (as yet unstated) reason for thinking that sensitivity to numerical precision is necessary to represent number. Since the number of whole items in a given set is expressed by a positive integer, we conclude that, at a minimum, the ANS represents positive integers
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