Cognitive scientists have argued that cognition is fundamentally embodied and that concepts are understood through perceptual simulation (Pecher & Zwaan, 2005; Semin & Smith, 2008). This also seems to hold true for numerical information, with participants processing small numbers faster with their left hand and large numbers faster with their right, akin to perceptually simulating numbers on a mental number line. This finding is known as the spatial–numerical association of response codes (SNARC; Dehaene, Bossini, & Giraux, 1993; Fischer & Brugger, 2011; Restle, 1970).

The SNARC effect is robust, with physical manipulations (e.g., crossing hands, grasping) and handedness failing to influence its direction (Andres, Ostry, Nicol, & Paus, 2008; Dehaene et al., 1993). SNARC holds for two-digit numbers (Dehaene et al., 1993; Reynvoet & Brysbaert, 1999) and number words (Fias, 2001), and extends to other ordinal-sequence-based organizational systems, such as alphabets, and large/small object words (Gevers, Reynvoet, & Fias, 2003; Ren, Nicholls, Ma, & Chen, 2011; Shaki & Gevers, 2011).

Several theories have been proposed to explain SNARC in terms of embodied cognition. Dehaene et al. (1993) suggested that numbers are spatially organized on a mental number line according to magnitude. Alternatively, SNARC might be an embodied association between numbers and actions (e.g., common patterns of motor activation are based on the left side of a keyboard having small numbers and the right side having large numbers (Gevers, Caessens, & Fias, 2005). Fischer and Brugger (2011) suggested that finger counting may be the origin of the effect. These theories share the idea that the SNARC effect is the consequence of embodied mechanisms.

However, some findings have questioned an embodied account. For instance, the original task demonstrated vertical (Ito & Hatta, 2004) and horizontal (Shaki, Fischer, & Petrusic, 2009; Zebian, 2005) effects, indicating that the mental number line is not canonical. In addition, illiterate participants fail to show a SNARC effect (Zebian, 2005). Moreover, Fischer, Shaki, and Cruise (2009) found that spatial representation is not inherent in numbers, but caused by directional reading conventions. These findings suggest that embodied mechanisms might not be the only explanation for SNARC and hint at a linguistic explanation.

Several psycholinguistic theories have suggested that cognitive processes are explained by both embodied and linguistic mechanisms (Louwerse, 2008), with experimental findings that have been attributed to perceptual simulation being alternatively explained by language statistics (Louwerse, 2008; Louwerse & Jeuniaux, 2010). Louwerse (2008) argued that language is organized so that it reflects embodied relations. That is, the prelinguistic conceptual knowledge (e.g., number magnitude) used when speakers formulate utterances gets translated into linguistic conceptualizations (language statistics) so that, as a function of language use, embodied relations are encoded in language. The fact that findings originally attributed to embodied cognition can also be attributed to language statistics begets the question of whether SNARC might be attributed to linguistic factors.

We compared embodied and linguistic accounts as possible explanations for SNARC. Two experiments replicated the SNARC experiments with number words (Dehaene et al., 1993; Fias, 2001; Nuerk, Iversen, & Willmes, 2004; Exp. 1 below) and Arabic numbers (Dehaene et al., 1993; Exp. 2 below). We predicted a strong negative correlation between number magnitude and number (word) frequency, as more-frequent numbers are also smaller (e.g., 1 is more frequent than 2), with both accounts explaining response times (RTs). We also investigated whether the collocation frequency of trial pairs could explain RTs, as this effect cannot readily be accounted for by embodied cognition. In general, evidence that word frequency elicits a SNARC effect strengthens the hypothesis that SNARC could be explained by language statistics.

Experiment 1

In Experiment 1, we determined whether SNARC is explained by number magnitude or by language statistics. If the SNARC effect has a linguistic basis, we should at least find it with number words (cf. Fias, 2001).

Method

Participants

A group of 57 right-handed native English-speaking undergraduate students participated for extra credit. Following Dehaene et al. (1993), in randomly assigned conditions participants were instructed to first respond to even numbers with their left hand and odd numbers with their right hand (n = 27), or to use the reverse mappings (n = 30).

Stimuli

Each experiment consisted of 65 trials, with each trial including two number words, ranging from one to nine (excluding five; Tzelgov, Meyer, & Henik, 1992).

Procedure

Number words were presented in the center of an 800 × 600 screen in 36-pt. font and subtended at most 2.5º of vertical visual angle from 60 cm. Two words were presented in each trial, but the words appeared on the screen one at a time. Participants were asked to determine number parity. After a participant responded to the first word, the second was presented. Although participants responded to all of the words, only RTs to the second word in each trial were analyzed. The stimuli within each trial were paired so that participants saw each number paired with every other number, in both orders (e.g., participants saw trials of both one followed by three and three followed by one). This allowed for all word pair frequencies to be accounted for. Once a participant had responded to both words in a trial, the next trial would commence after a short beep and after the “+” symbol had appeared for 1,000 ms. Every trial was so separated as to provide space between the trials. Trial pairs were randomly presented, and participants saw every combination of pairs. Six practice trials preceded the experiment.

Results and discussion

Five participants were removed because >14 % of their answers were incorrect. The average error rate was 5 %. Outliers were identified as responses faster than 200 ms or slower than 1,500 ms, following the criteria of Shaki et al. (2009). Errors and outliers were removed, affecting 6.5 % of the data.

As in Dehaene et al. (1993) and Fias (2001), the median RT per number word per response side was separately computed per participant.Footnote 1 Median left-hand responses were subtracted from median right-hand responses. A mixed-effects regression was conducted on RTs, with response side and magnitude as fixed predictors and participant and item as random predictors. The model was fitted using restricted maximum likelihood estimation (REML) for the continuous variable (RT). The F-test denominator degrees of freedom were estimated using Kenward–Roger’s degrees-of-freedom adjustment, in order to reduce the chance of Type I error (Littell, Stroup, & Freund, 2002). Evidence supporting an embodiment account stems from the interaction between faster left-hand responses for smaller numbers and faster right-hand responses for larger numbers, as this interaction links space (right/left hand) and magnitude. A second regression with response side and linguistic frequency as fixed predictors was also performed. The linguistic factor was operationalized as the log frequency of the number word (see Table 1). This frequency was obtained from the Web IT one-trillion-word 5-gram corpus (one trillion word tokens, with 13,588,391 word types from 95,119,665,584 sentences; Brants & Franz, 2006). Log frequency is typically preferred over raw frequency because the distribution of word frequency is right-skewed (i.e., L-shaped) (Baayen, 2001).

Table 1 Bigram and unigram log frequencies of number words for Experiment 1
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A main effect emerged for response side, with faster RTs for right-hand responses, F(1, 5815.85) = 6.57, p = .01, R 2 = .10. This result is not surprising, with all participants being right-handed. More interestingly, we found an interaction between response side and magnitude, F(1, 5816.93) = 3.26, p = .04, R 2 = .04 (Fig. 1), replicating a SNARC effect.

Fig. 1
figure 1

Linear fitting of the SNARC effect for Experiment 1 (number words)

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As predicted, we found a strong negative correlation between magnitude and frequency, r = −.98, p < .001 (cf. Dehaene & Mehler, 1992). This allows for the possibility that SNARC can be explained by word frequencies. Frequency should then not affect RTs, but an interaction is predicted between response side and frequency. The SNARC effect predicts small numbers to be processed faster with the left hand; if word frequency alone affected RTs, we would expect faster processing of frequent words regardless of response side. We found that frequency itself did not explain the RTs, F(1, 5587.95) = .01, p = .93, R 2 = .0003, but, analogous to the SNARC effect, an interaction was apparent between response side and frequency, F(1, 5586.16) = 3.23, p = .04, R 2 = .04 (Fig. 2).

Fig. 2
figure 2

Linear fitting of the statistical linguistic frequencies for Experiment 1 (number words)

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Whether the linguistic system simply provides redundant information derived from the perceptual system is still unanswered, because what is explained by frequency is also explained by magnitude. To test whether linguistic frequencies independently explain the findings, we analyzed the collocation frequencies of paired number words in each trial (see Table 1). If statistical linguistic frequencies of the word pairs explain RTs, this finding would be difficult to attribute to embodied mechanisms because collocation frequencies cannot be explained by the magnitude of the second word. No correlation emerged between collocation frequencies and the second number’s magnitude, r = −.15, p = .20. In a mixed-effects model, bigram frequency significantly explained RTs of the second word in each pair, F(1, 3072.72) = 4.12, p = .04, R 2 = .14, with higher frequencies yielding lower RTs. A significant interaction was found between response side and frequency, F(2, 3082.32) = 3.54, p = .03, R 2 = .12. These collocation results thus mirror the traditional SNARC findings, but they are difficult to explain with an embodied account, providing evidence for a linguistic account.

Experiment 1 demonstrated that language statistics explain RTs as well as an embodied cognition account (SNARC) does. Moreover, the RTs could be explained by language statistics but not by an embodied cognition account. However, the argument could be made that Experiment 1 used number words and therefore was biased toward a linguistic account.

Experiment 2

In Experiment 2, we used Arabic numerals instead of number words, as Arabic numerals may be processed differently from number words (Damian, 2004). We also included 0, whose low magnitude, yet lower frequency than other low-magnitude digits, allowed for comparing an embodied account (that magnitude explains SNARC) and a frequency account (that frequency explains SNARC) (cf. Pinhas & Tzelgov, 2012). In other words, the number 0 appears less frequently than the number 1, yet its magnitude is less than 1.

Methods

Participants

A group of 44 right-handed native English speaking undergraduates participated for extra credit. The participants were evenly split between response conditions.

Stimuli and procedure

Each experiment had 81 trials, including two Arabic numerals presented one at a time, ranging from 0 to 9 (excluding 5). The procedure, font size, and viewing angle were identical to those in Experiment 1. Participants were again asked to determine number parity, with instructions that specified that 0 was an even number.

Results and discussion

Eight participants were removed because >14 % of their answers were incorrect. A software error led to the loss of 2.2 % of the data. Outlier removal resulted in data loss of 2.43 %.

Our analysis was the same as in Experiment 1, in which median RTs per number word per response side were separately computed for each participant (see also the table in note 1). Response side explained RTs, F(1, 4973) = 1.20, p < .001, R 2 = .02, and the interaction between response side and magnitude was significant, F(1, 4973) = 13.88, p < .001, R 2 = .23 (Fig. 3), replicating the SNARC effect.

Fig. 3
figure 3

Linear fitting of the SNARC effect for Experiment 2 (Arabic numerals)

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Similar to the negative correlation between number words and magnitude in Experiment 1, we found a negative correlation between Arabic numerals and their frequencies, r = −.60, p < .001. Note that the correlation was weaker than before, because we included 0. Without 0, the correlation was stronger, r = −.98, p < .001. Frequency did not affect RTs, F(1, 4973) = 0.05, p = .81, R 2 = .001, but the Response Side × Frequency interaction was significant, F(1, 4973) = 14.60, p < .001, R 2 = .24 (Fig. 4).Footnote 2 This finding replicated the SNARC effect and is similar to the results of Experiment 1, except that it was now obtained with numbers rather than number words.

Fig. 4
figure 4

Linear fitting of the statistical linguistic frequencies for Experiment 2 (Arabic numerals)

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As before, we assessed frequency collocations for pairs to determine whether bigram frequency alone impacted RTs. Bigram frequencies did not correlate with the magnitude of the second word in the pair, r = .08. A main effect of response side was found, F(1, 2098) = 9.29, p < .01, R 2 = .12, and bigram frequency did not significantly explain RTs, F(1, 2098) = .03, p = .88, R 2 = .001. Importantly, the Response Side × Frequency interaction was significant, F(1, 2098) = 42.22, p < .001, R 2 = .53, a finding that cannot be explained by an embodied account. See Table 2 for the bigram and unigram log frequencies of the Arabic numerals.

Table 2 Bigram and unigram log frequencies of Arabic numerals for Experiment 2
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Including 0 allowed us to compare the two accounts, because 0 has the lowest mathematical and psychological magnitude (Pinhas & Tzelgov, 2012), yet it has a lower frequency than the other low-magnitude numbers. Left-hand responses for 0 were slower (M = 670 ms) than right-hand responses (M = 641 ms), albeit not significantly, t(555.65) = −1.5, p = .13. To determine whether the RT findings for 0 provided support for a frequency or embodied account, we compared the RTs for the items 0 and 1. If magnitude explained responses, because both numbers shared low magnitudes, no significant difference was expected between them. But if word frequency explained the responses, because 1 is quite frequent and 0 is less frequent, the RTs for these two items were predicted to be divergent, which was what we found, t(10.75) = −4.5, p < .001. However, the differences between 0 and 1 might be explained by a learned embodied relationship, with the “0” key on a keyboard being on the right. To support such an explanation, it would be necessary for RTs to 0 to be faster with the right hand, but they were not, t(555.65) = −1.5, p = .13.

Experiments 1 and 2 demonstrated a language statistics explanation for the SNARC effect. This evidence does not replace the SNARC effect, because we did find an effect of magnitude. In fact, in Experiments 1 and 2, participants seemed to use frequency information when making parity judgments about either number words (Exp. 1) or Arabic numerals (Exp. 2), as was evidenced by the significant Magnitude × Response Side and Frequency × Response Side interactions. Although Damian (2004) claimed that number words and Arabic numerals are processed differently, such that when processing Arabic numerals information about magnitude is more readily available, and when processing number words, lexical information is more readily available, participants in Experiments 1 and 2 were asked to make parity judgments, calling explicit attention to neither magnitude nor frequency.

With the findings from Experiments 1 and 2, at least two arguments favor a frequency account. First, when 0 was included—a number with both low magnitude and lower frequency—no SNARC effect was found, even though a frequency effect was obtained. Second, bigram frequencies explained the RTs, whereas such an explanation is lacking for an embodied account. However, a problem with a language statistics account concerns direction. Whereas number-line representations explain why left-hand responses are faster for low-magnitude items, the rationale is not so obvious for high-frequency numbers eliciting faster left-hand responses.

Markedness can explain this pattern. Greenberg (1966) argued that for any word pair, the one that is more frequent is the unmarked (i.e., most natural, simplest, first learned), and the one that is less frequent is the marked member of the pair, with unmarked members preceding marked members (Louwerse, 2008). Although this explanation seems similar to Proctor and Cho’s (2006) polarity correspondence principle, a markedness explanation suggests that for any given pair, items will be processed faster when frequent items appear before infrequent items, not when items are matched (to their response sides) on polarity. This general linguistic theory of markedness ranges over phonological, grammatical, and semantic elements (Chomsky & Halle, 1968) and can be applied to number words, with frequent items being processed faster with the left hand. The bigram frequencies for pairs of number words in the Web IT one-trillion-word 5-gram corpus (Brants & Franz, 2006) show that frequent number words precede infrequent number words more often than vice versa, F(1, 70) = 31.25, p < .001. If the frequency asymmetry is an explanation for the direction of the language statistics effect found in Experiments 1 and 2, then we would predict a “SNARC” effect, with high-frequency (nonmagnitude) words being processed faster with the left hand, and vice versa for the right. This hypothesis was tested in Experiment 3.

Experiment 3

Method

Participants

A group of 49 students participated for extra credit. Of these participants, 22 were randomly assigned to first respond to animate words with their left hand and inanimate words with their right hand, and 27 to the opposite mapping.

Stimuli

In all, 30 two-word trials were presented one word at a time. The words extracted from the MRC Psycholinguistic Database were frequent or infrequent and were matched on word length (see Table 3). The word frequencies of frequent and infrequent words differed significantly, t(69) = −17.10, p < .001. Half of the words described animate concepts, whereas the rest described inanimate concepts.

Table 3 Unigram log frequencies of the experimental stimuli for Experiment 3
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Procedure

The procedure, size, and viewing angle were identical to those in Experiments 1 and 2. Participants were asked to indicate whether a word that appeared on the screen represented something animate or inanimate.

Results and discussion

Seven participants were removed because >14 % of their answers were incorrect. Outlier removal resulted in the loss of 2.1 % of the data.

Median RTs per word per response side were separately computed for each participant (see also note 1). The same mixed-effects regression was conducted on RTs as in the previous experiments. Response side significantly predicted RTs, F(1, 1856) = 3.73, p = .05, R 2 = .14, with right-side responses being faster. Frequency approached significance, F(1, 1856) = 3.24, p = .07, R 2 = .12. Importantly, the Response Side × Frequency interaction was significant, F(1, 1856) = 7.23, p < .01, R 2 = .27, indicating that high-frequency words were indeed processed faster with the left hand, whereas low-frequency words were processed faster with the right Fig. 5.

Fig. 5
figure 5

Linear fitting of the statistical linguistic frequencies for Experiment 3 (high- and low-frequency words)

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General discussion

The SNARC effect adds to the large body of literature suggesting that cognition is fundamentally embodied. Yet several studies have demonstrated that language statistics can explain the experimental findings equally well (Louwerse, 2008; Louwerse & Jeuniaux, 2010). The present results add to these findings. Not only does a frequency account explain the findings, collocation frequencies were able to explain the RTs, where an embodiment account could not. Also, RTs from the number 0, a number with both low magnitude and frequency, supported the frequency and not the embodiment account. Finally, a SNARC-like effect was found for low- and high-frequency words, for which embodiment could not be the explanation.

The finding that linguistic frequencies explain SNARC does not dismiss an embodiment account. After all, the interaction of response side and word frequency could be considered embodied. However, the source of SNARC is not necessarily magnitude on a perceptually simulated mental number line. Complementary to this explanation, it can be argued that language has encoded such perceptual information, so that language users rely on language statistics in their cognitive processes (Louwerse, 2011). Consequently, frequency would then be likely to explain SNARC-like effects obtained with a variety of stimuli, whether magnitude information was present or not, such as with ordinal information or even large/small object words (Gevers et al., 2003; Ren et al., 2011; Shaki & Gevers, 2011).

In addition, pivotal work on SNARC has found reading direction to be a relevant factor. In the present work, it is certainly difficult to tease apart one component of language (a linguistic frequency effect) from another (a particular reading direction). Future work needs to determine whether this effect holds with languages that have different reading directions, such that frequent/unmarked forms are processed faster when they precede infrequent/marked forms, regardless of directional reading conventions; for example, when reading from right to left, would frequent words instead be processed faster with the right hand?

The notion of frequencies playing a role in numerical cognition is not new. Dehaene et al. (1993) evaluated the interactions between number and word representations and showed that treating them as eliciting separate processes is not an accurate description of number processing. This conclusion is reminiscent of the conclusion drawn by Louwerse (2008) that the nature of conceptual processing is symbolic and embodied. Language statistics facilitate cognitive processes because language encodes magnitude.