On What Ground Do Thin Objects Exist? In Search of the Cognitive Foundation of Number Concepts

Abstract

Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations.

An empirically informed account of numbers as reifications

“…Estimation is the ability to assess the size of larger collections approximately, with increasing error (the larger the collection, the larger the estimation error) (Dehaene, 2011;Gilmore et al, 2018;Knops, 2020). Although these abilities are often called 'numerical' in the literature, they are not really numerical, since they do not display the most distinguishing properties of numerical systems (dos Santos, 2022;2023;Núñez, 2017). For example, subitising has a very low upper limit (whereas numbers constitute an infinite set), and estimation is imprecise and produces only fuzzy estimates (whereas numbers are exact).…”

“…For example, subitising has a very low upper limit (whereas numbers constitute an infinite set), and estimation is imprecise and produces only fuzzy estimates (whereas numbers are exact). Núñez (2017) coined the neologism 'quantical cognition' to distinguish these abilities from proper numerical cognition, and Pantsar (2014Pantsar ( , 2023 characterises them as 'proto-arithmetical', highlighting the observation that these abilities, although non-numerical, constitute genetically evolved preconditions for arithmetic.…”

On Radical Enactivist Accounts of Arithmetical Cognition

“…While platonist views are still entertained in contemporary philosophy of mathematics (e.g., S. Shapiro 2007;Brown 2008), they are seen as increasingly difficult both ontologically and epistemologically (see, e.g., Benacerraf 1973;Linnebo 2018;Pantsar 2021b;2021c). The main ontological problem involves the status of mathematical objects (or structures), which-being non-causal, nonspatial and non-temporal-would be unlike that of any other objects.…”

From Maximal Intersubjectivity to Objectivity: An Argument from the Development of Arithmetical Cognition

“…It should be added here that nothing in the present account demands assuming the existence of mind-independent mathematical objects. See(Pantsar 2021a) for an argument on how mathematical objectivity does not demand mathematical objects and(Pantsar 2021d) for more on the existence of mathematical objects.12 This paper was researched and written during my period as a Senior Research Fellow at the Käte Hamburger Kolleg: Cultures of Research, RWTH Aachen University, Germany, funded by the German Federal Ministry of Education and Research. I would like to thank Regina Fabry for discussions on the themes of this paper, as well as the two anonymous reviewers for helpful comments.…”