Whatever Be Their Number: Counting on the Visible, the Audible, and the Tangible

Introduction

Four-year-old Kai is counting small chocolate Easter eggs that he has unearthed and carefully moves them one at a time from the distal pile of the as-yet-uncounted to the proximal pile of the already-counted: “…, nine, ten, eleven, twelve, thirteen”. He holds up the next-to-be-counted and asks, “What’s this one called?” and is told, “Fourteen”. “And this one?” he asks, holding up his next selection. “Fifteen.” And then he is off again, both counting and counting eggs in concert all the way to twenty. At a significant level, to count is to name; to be able to count is to know the number names and to be able to recite them correctly, in order. And naming brings number into being: in English, to tell means both to say and to count.

Pimm (1995, pp. 64-66) distinguishes between what he terms transitive and intransitive counting. By means of these terms, he signals connections on the one hand between counting things and ‘counting’ as a verb with a direct object (transitive), and on the other between ‘just counting’ (reciting the number names in order) and ‘counting’ as a verb with no direct object (intransitive), where no things are actually being counted (except perhaps the number words themselves). Kai is well on his way to mastering two interlocking but distinct cultural practices (those of transitive and intransitive counting). In asking ‘What’s this one called?’, he reveals a sense of transitive counting as invoking a temporary sequential naming that feels far more ordinal than cardinal in nature. However, the actual sleight-of-hand of transitive counting occurs at the end of the procedure, when a specific number word is re-assigned to refer to the whole collection (in answer to the question “How many?”), one which matches (promotes? copies? usurps? steals?) the name already allocated to the last-to-be-counted¹. Might we not just as well – and perhaps more precisely – count things using ordinal names: first, second, third, fourth, …, where there would at least be a clearer linguistic shift from the terminal point of an ordinal practice to a (nonetheless related) cardinal word?

The ordinal and cardinal systems in English are very close linguistically: with the exception of four of the first five words of each (first to fifth, one to five), you can move from ordinal to cardinal by removing ‘-th’ (or sometimes ‘h’) or, conversely, move from cardinal to ordinal by means of adding it². In French, it is very similar (where the ordinal suffix is ‘-ième’ and the system becomes regular more quickly); in German, the suffix is ‘-te’ (and is intermittently regular among the first few before hitting its stride), and so on³. In relation to Ancient Greece, Fowler (1987) refers to arithmoi as cardinals, but helpfully observes:

Ordinals convey a sense of time and sequence, of ‘the next one to be named’ and ‘the one to be said after that’. As mentioned above, success with intransitive counting primarily involves being able every time to generate stably the same set of words in the same order⁴. Transitive counting, which may be over-emphasised in early schooling (see Tahta, 1991), relies centrally upon intransitive counting and can actually be seen as a ‘mere’ application of it, a subordinate practice. And as Seidenberg (1962) observes, with regard to counting’s possible origins, “an application of a device (or idea) is an effect of a device, not a cause” (p. 2). He believes transitive counting, the counting of things, is most unlikely to be the origin of counting.