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 (metaphysically) 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 1 future empirical research.With just a little reflection it is clear that there are two roles which the natural numbers play. First of all, they can function as cardinal numbers. The natural numbers can be used to specify the number of objects falling under a concept. This is the role in which they appear when a number is given in answer to the question 'how many Fs are there?'. Second, the natural numbers can function as ordinal numbers. In this case they specify the position of an item in an ordering, for example, to say that a particular person finished third. If one wants to be even more explicit that the same number is used in such a case, consider that the same thing can be said with 'this person ended up being number three'. Rather than giving the size of a set, ordinal numbers give the position, and so answer questions of the form 'which F is this?'.The dual role of the natural numbers has long been known, just as it has long been known that it is possible to define cardinal numbers in terms of ordinal numbers. Cantor (1883) introduced the necessary machinery for defining a cardinal number based on an ordinal number in a well-ordering. According to Cantor (1895) the 'general concept' of a cardinal number, "arises from [a set] M when we make abstraction of the nature of its various elements m and of the order in which they are given [...] Since every single element m, if we abstract from its nature,

Replies

“…This claim has spurred a fair amount of discussion; see, e.g.,Snyder et al (2018),Assadian & Buijsman (2019), andBuijsman (2021).REPLIES…”