Can the principles underlying the structure of natural numbers be inferred from interactions with the physical world? According to Rips et al., the transition from sensorimotor experience to mathematical concepts is largely undetermined and cannot account for the acquisition of natural numbers. We argue that fingers may provide the missing tool to apprehend numbers in the physical world, and we put forward striking similarities between finger-counting strategies and the principles underlying the structure of natural numbers.

Children spontaneously use a stable sequence of finger movements while counting, presumably due to the joint influence of motor constraints and cultural habits. The stable-order principle emerging from the use of consistent finger-counting strategies involves the understanding that natural numbers include a unique first element (e.g., thumb or little finger, depending on cultural habits), a unique immediate successor for each element in the sequence (e.g., from thumb to little finger, there is only one successor for each finger), and a unique immediate predecessor for each element except the first (e.g., from index to little finger, there is only one preceding finger). Finger-based representations therefore preserve cardinality and maintain a one-to-one correspondence between the physical world and symbolic systems.

Moreover, once the sequence of numbers from 1 to 10 is instantiated in finger counting, fingers can be used to keep track of the successors of any other number, providing an empirical ground for mathematical induction. Finger counting goes even farther, as it allows the children to infer the base-10 mathematical system. Historically, the base 10 has been the most widely used, as evidenced by the traces left by Amorites, Incas, and Tibetans (Ifrah Reference Ifrah1981). There is no reason to explain this choice relative to others, such as the more convenient base 12, which has more divisors, except that the base-10 system has a physical counterpart in finger use.

The influence of finger-counting strategies on number representations is supported by several empirical facts. A first evidence comes from the origin of number words in many languages (e.g., in English, five comes from a common root of finger and fist, digit means at the same time number and finger; Menninger Reference Menninger1969). This embodied vocabulary suggests that counting originates from the use of fingers rather than using arbitrary quantitative words. Developmental studies showed that the score obtained in finger discrimination tasks is the best predictor of arithmetical performance in children (Fayol et al. Reference Fayol, Barrouillet and Marinthe1998; Noël Reference Noël2005). Moreover, the errors made by children in mental calculation often reveal a 5-unit difference relative to the correct response (e.g., 18–7=6), which is reminiscent of the representation of intermediary results on a full-opened hand (Dohmas et al. Reference Dohmas, Krinzinger and Willmes2008). Furthermore, training children with poor mathematical achievement to discriminate their fingers was found to improve both their finger and number knowledge (Gracia-Bafalluy & Noël Reference Gracia-Bafalluy and Noël2008).

The question arises as to whether finger-counting strategies still influence numerical cognition in adults. To address this issue, we conducted an experiment in which participants had to identify Arabic digits by pressing the keyboard with a different finger for each number. Results showed that responses were faster and more accurate when the finger assigned to each number matched the finger-counting strategy of the participants (Di Luca et al. Reference Di Luca, Grana, Semenza, Seron and Pesenti2006). In another experiment, we showed that number naming was faster when participants were presented with canonical finger configurations, that is, configurations conforming to their personal finger-counting habits, rather than non-canonical ones. Moreover, we observed that comparing Arabic digits was faster when the digit display was preceded by the unconscious presentation of either canonical or non-canonical finger configurations. However, only the priming effect induced by canonical configurations generalized to new, never consciously seen, numerosities, which provides clear evidence for an automatic access to semantic numerical knowledge from finger-counting configurations (Di Luca & Pesenti Reference Di Luca and Pesenti2008). Finally, we showed that canonical configurations are processed as symbolic systems and activate a place-coding semantic representation of numbers, whereas non-canonical configurations activate a summation-coding semantic representation (Di Luca Reference Di Luca2008; Di Luca & Pesenti, under revision; Roggeman et al. Reference Roggeman, Verguts and Fias2007).

The hypothesis that knowledge of natural numbers is grounded in finger representations is also supported by neuroimaging data showing that, in numerical tasks, activation was observed in the parieto-precentral circuits classically associated with finger movements (Pesenti et al. Reference Pesenti, Thioux, Seron and De Volder2000). Electrophysiological results converge to the same conclusion (Andres et al. Reference Andres, Seron and Olivier2007; in press). For example, Sato et al. (Reference Sato, Cattaneo, Rizzolatti and Gallese2007) found that the amplitude of muscle twitches induced by transcranial magnetic stimulation (TMS) in the right hand increased when participants performed a verbal parity judgment on numbers 1–4 relative to numbers 6–9. Because all participants reported starting counting with their right hand, this result could reflect a specific contribution of hand motor circuits to number processing.

Although these findings favor a unique relationship between numbers and fingers, Rips et al. may still argue that finger counting is mediated by innate and abstract representations of natural numbers. However, the assumption that natural numbers are preconfigured is contradicted by the finding that some Amazonian populations do not develop the complete number sequence (Pica et al. Reference Pica, Lemer, Izard and Dehaene2004). Interestingly, this limited numerical knowledge was found to co-occur with a rudimentary finger-counting strategy, which supports the idea that finger counting may critically contribute to understanding natural numbers. Accordingly, neuropsychological studies have shown a joint deficit in calculation and finger discrimination following a vascular damage (Gerstmann Reference Gerstmann1940) or a virtual TMS lesion of the parietal cortex (Rusconi et al. Reference Rusconi, Walsh and Butterworth2005).

In conclusion, we showed that finger-counting strategies are sufficient to allow the development of numerical representations that integrate the basic properties of natural numbers as well as higher-order properties such as the base-10 concept. We therefore argue that it is not necessary to presuppose an abstract schema specifying the structure of natural numbers to explain the development of the concept of numbers. The finding that finger-counting strategies still influence number processing in adults rather suggests that knowledge of natural numbers may build up on finger-based representations through a bottom-up process.