An empirically feasible approach to the epistemology of arithmetic

Abstract

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. KeywordsArithmetical cognition • Philosophy of mathematics • Epistemology • Empirical study • Contextual a priori 1 1 Introduction: a wish-list for an epistemological theoryIn this paper I wish to propose a framework for an empirically feasible epistemological theory of arithmetic. The bulk of the paper focuses on explaining what that empirical feasibility entails, but we should first establish the context by specifying what we generally desire of an epistemological theory of mathematics. While there is bound to be little consensus over any comprehensive and detailed list of criteria, it seems that

The Modal Status of Contextually A Priori Arithmetical Truths

“…In Pantsar (2014), I have proposed an empirically feasible approach to the epistemology of arithmetic. The purpose of the paper is to study what kind of epistemological theory is required if the recent results in neurobiology, cognitive science and psychology are accepted as giving insight to the origins and nature of arithmetical knowledge.…”

“…The whole point of approaches like Pantsar (2014) is that we do not need to evoke mind-independent existence for numbers in order to have feasible epistemology of arithmetic. But at the same time, the theory is based on biological primitives and as such entails a high level of objectivity for arithmetical truths.…”

“…In this paper I will study these questions with regard to the contextual a priori theory of epistemology arithmetic proposed in Pantsar (2014). In that account, arithmetical knowledge is thought to be empirically constrained by biological primitives that we have as infants and share with many animals.…”

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AbstractIn Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character.

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“…The proto-arithmetical cognitive system enables understanding that numerals one, two, three, and four refer to dierent numerosities that form a succession. In learning to count, we generalize on this process, that is, 3 The empirical details and references can be found in Pantsar (2014). For an introduction to the empirical studies, see Dehaene (2011) and Dehaene & Brannon (eds.)…”