Two roads to the successor axiom
Stefan Buijsman 0
0 Filosofiska Institutionen, Stockholm University , Universitetsvägen 10D, 106 91 Stockholm , Sweden
Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don't claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children's responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is an empirically viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers.
Successor axiom; Epistemology; Number concepts; Arithmetical cognition
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My thanks to three anonymous reviewers for their extensive comments on earlier versions.
1 Introduction
An important question for any philosophical account focussed on basic arithmetic is
how we learn that the natural numbers ‘keep going’, i.e. that for every natural number
there is another, larger, natural number. This aspect of the natural numbers is codified
in the successor axiom of Peano arithmetic (with Nx meaning ‘x is a natural number’
and x P y meaning that x is the predecessor of y):
(Successor) ∀x (Nx → ∃y x P y)
While this is a difficult question for any account, my main interest here will be to
see how we might answer it on the basis of what actually happens in childhood. In
other words, how is it that virtually everyone manages to acquire number concepts
in childhood that are consistent with the successor axiom? In order to answer that
question, it is important to not only look at philosophical accounts, but to connect
these to the empirical data that is available. My aim is to provide a philosophical
account of our knowledge of the successor axiom that is at the same time empirically
feasible. So, to provide an account that is in accordance with what we know about
actual child development. In doing so I will not provide an account that is specific to
platonism or nominalism, as I suspect that the underlying data can be interpreted in
both directions. I will also not provide a more general account of our knowledge of
actual infinity, which has been the aim of e.g.
Lakoff and Nuñez (2000)
and
Pantsar
(2015)
. I only intend to provide an account of how we can acquire number concepts
that match the successor axiom. In other words, to explain how children figure out
that the numbers ‘go on forever’, so that there are infinitely many numbers. I will talk
of ‘matching’ or ‘being consistent with’ the successor axiom as a way to state more
precisely that children have figured out that there are infinitely many numbers.
My aim is to offer an account of children’s knowledge that is based on the structure
of the numeral system and not on the procedure of adding one. I argue that such an
account is a viable alternative to the existing (plus-one) account, even though it is
not yet possible to decide between the two. First, I lay some of the groundwork for
my own position by discussing what we know about our grasp of numbers that are
represented by more than one digit (e.g. 84). That provides a basis of information about
the mechanisms that are used to understand the structure of our numeral system, which
I claim can explain how we acquire concepts that are consistent with the successor
axiom. The work in Sect. 2 thus prepares the different elements for this account by
discussing the structural aspects of the numeral system and how children learn these
aspects. It is particularly relevant for Sect. 4, where the aspects of the numeral system
identified in Sect. 2 is used to interpret parts of the empirical data.
Before working out my own account I briefly discuss the kind of account that
has been prevalent in the philosophical literature. This is the alternative explanation
of how we acquire concepts consistent with the successor axiom. Philosophers such
as
Parsons (2007)
have claimed that we can acquire knowledge on the basis of the
successor relation. I do not criticise the possibility of such an approach, but do claim
that this is not the only currently viable hypothesis. The alternative is (...truncated)