The representations of the approximate number system

ABSTRACT

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements.

KEYWORDS:

• Approximate number system
• mental representations
• philosophy of arithmetic
• number cognition

Acknowledgments

My thanks to Øystein Linnebo, Fernando Marmolejo-Ramos, Peter Pagin, Marco Panza, Dag Westerståhl, and the anonymous reviewers for comments on previous versions of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. To give two examples, the ANS counts as having representations both on the account of Gallistel and that of Burge. It meets the criterion put forward by Gallistel: “The brain is said to represent an aspect of the environment when there is a functioning isomorphism between some aspect of the environment and a brain process that adapts the animal’s behaviour to it” (Gallistel, 1990, p. 15). It also meets the criterion put forward by Burge: “According to Burge, perceptual constancies enable individuals to distinguish what is happening at their surfaces from what is happening in the world, thereby giving rise to non-trivial veridicality conditions” (Beck, 2015, p. 843).

2. A lot of the experiments use collections of dots on a screen or piece of paper – hence this particular example.

3. Recall that I do not intend to cover in my account how this happens, that is, I remain neutral on the underlying psychological mechanisms. Theories of those mechanisms, such as those by Gebuis et al. (2016) and Dehaene et al. (2003), do talk about the abstraction procedure. I only focus on the narrow question of what the existence of such an abstraction procedure means for the content/vehicle of the representations of the ANS.

Additional information

Funding

This work was supported by Vetenskapsrådet [Grant no. 2018-01163].

Notes on contributors

Stefan Buijsman

Stefan Buijsman (1995) completed his PhD at Stockholm University on the philosophy of mathematics, with a focus on the use of empirical findings regarding mathematical cognition to test philosophical accounts of mathematics. He has continued that work since and now works as a researcher at the Institute for Futures Studies on a Vetenskapsrådet grant on the relevance of empirical results for the philosophy of mathematics.