Are the Natural Numbers Fundamentally Ordinals?

Abstract

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well‐known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as fundamental. By discussing structuralism and neo‐logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research.

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Bahram Assadian
University of Amsterdam

References found in this work

Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford, England: Oxford University Press.
The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.
Introduction to Mathematical Philosophy.Bertrand Russell - 1919 - Revue Philosophique de la France Et de l'Etranger 89:465-466.
Core Systems of Number.Stanislas Dehaene, Elizabeth Spelke & Lisa Feigenson - 2004 - Trends in Cognitive Sciences 8 (7):307-314.

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Citations of this work

Abstractionism and Mathematical Singular Reference.Bahram Assadian - 2019 - Philosophia Mathematica 27 (2):177-198.

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