Two roads to the successor axiom

Abstract

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.

Acquiring mathematical concepts: The viability of hypothesis testing

“…On a concept externalist framework this means that children meet, at a minimum, the attribution conditions of the number concepts. However, if one insists on pressing the point, then I have argued elsewhere (Buijsman, 2019b) that children (can) learn that there are infinitely many numbers on the basis of the syntax of numerals. The place‐value system offers a clear, purely syntactic, method for constructing numerals that refer to larger numbers.…”

Building blocks for a cognitive science-led epistemology of arithmetic

“…In short, while there have been a number of attempts to formulate an epistemology of arithmetic based on results from cognitive science, they run into trouble because they rely exclusively on the OTS and ANS. I will formulate an alternative in the rest of this paper that tries to avoid this shortcoming, relying on earlier work of mine on the acquisition of number concepts (Buijsman, 2019(Buijsman, , 2020. Yet I will also maintain an approach similar to that of Burge (2007), dividing the epistemology into a part for small numbers (roughly those represented by singledigit Arabic numerals, though the OTS is more limited) and larger numbers.…”

Estimating sea floor dynamics in the Southern North Sea to improve bathymetric survey planning

“…Buijsman, 2007;Knaapen, 2005]. The stochastic character of surveyed depth values of the sea floor complicates 58 Chapter 3.…”

“…Dorst et al [2007]; Knaapen [2005]; Van Dijk and Kleinhans [2005]. A method called spatial cross-correlation McElroy et al, 2008;Buijsman, 2007;Duffy and Hughes-Clarke, 2005] has recently been used frequently to calculate migration rates, without using stochastic information on the quality of the depth observations. Methods for the analysis of sea floor dynamics are still in development, as a result of the recent improvements in description of the stochastic character of surveyed depth values of the sea floor.…”