An Input Lexicon for Familiar Numbers

When we talk about numbers, we mainly think about quantities. However, numbers may evoke different meanings: the cardinal value represents the size of a set, or an amount (e.g., “There are 900 places in the concert hall”), the ordinal value represents the place or the rank in a set (e.g., “The 900^th participant will win a boat trip”). Numbers may also refer to a nominal value, and be used as simple labels (Peugeot 5008), known phone numbers (911), pin codes, or evoke a famous event (1945). Our stored knowledge about numbers is referred to as “number facts”: we all know some arithmetic facts (e.g., 12 x 12 = 144) as well as encyclopedic number facts, like those just mentioned. Several case-studies in neuropsychology have suggested that encyclopedic number knowledge might be processed differently than arithmetic facts or unknown numbers with a cardinal or ordinal value (Cappelletti et al., 2008; Cipolotti, 1995; Cohen et al., 1994; Delazer & Girelli, 1997). In the current study, we address this issue in healthy participants, by assessing how encyclopedic numbers are read when they are visually presented. More specifically, we investigate if they generate reading processes that are similar to reading words rather than reading unknown numbers, and thus if they could be stored in a lexicon for Arabic digits by analogy to the orthographic lexicon for words.

Reading multidigit Arabic numbers is a complex cognitive operation, involving specific syntactic parsing processes on the input form, that are radically different from what happens when we read words (Dotan & Friedmann, 2018, 2019). Indeed, a multidigit Arabic number is constituted of a chain of 10 different possible characters (the digits 1-9 and 0), the value of which is defined by the position they hold in the sequence starting from the rightmost position and increasing by a power of ten at each step to the left. As an example, the digit “2” in 3472 means the quantity {2} x 10⁰, while in 3275, it means the quantity {2} x 10². When a power of 10 is not associated with any base quantity, then the digit 0 is used as a place-holder, e.g., in 3072. Finally, Arabic numbers are parsed into triplets from the rightmost position, also a striking difference with word reading that are read from left to right. These triplets are often marked in the visual array by a dot, a comma, or simply a space depending on cultural conventions (e.g., “thirty-four thousand seven hundred” is written “34.700”, “34,700” or “34 700”). Removing this triplets’ visual marker in very large numbers renders them almost impossible to read (e.g., 105975300 vs. 105.975.300). Reading aloud furthermore involves specific conversion processes in order to plan the corresponding verbal form, which corresponds to several words, rather than several phonemes as is the case in word reading. The appropriate number words, organized in lexical classes (units, decades or tens, and teens) and multiplier words (hundred, thousand, million, etc.), have to be retrieved in the correct order for them to represent the additive (e.g., “one hundred fifteen”: 100 + 15) or multiplicative (e.g., “fifteen hundred”: 15 x 100) relationships between lexical primitives.

Can this complex process be bypassed for encyclopedic numbers, so that they are recognized at a glance, as familiar words that are stored in an input orthographic lexicon? Some authors indeed suggested the existence of multiple routes for reading numbers (Cipolotti, 1995; Cohen et al., 1994), establishing parallels with the general cognitive architecture postulated in word reading (DRC model, Coltheart et al., 2001). The sublexical route, or grapheme-to-phoneme conversion route, would correspond to the transcoding process described above. Depending on the theoretical views, this route would mandatorily involve access to the representation of magnitude for numbers (McCloskey, 1992; McCloskey et al., 1985) or not (Barrouillet et al., 2004; Dehaene & Cohen, 1995; Power & Dal Martello, 1997; Seron & Deloche, 1984). The lexical route, if it exists, would imply the existence of an Arabic input lexicon where frequent and familiar number forms are stored (or “lexicalized”) and which addresses a phonological output lexicon, with or without access to semantics. Semantics in this case would be essentially non-quantity-related: it would hold properties and characteristics of these items at a conceptual level.

This multiple-route proposal for reading numbers is exactly what is suggested by the existence of several case-studies showing dissociations between the ability to read encyclopedic vs. unknown numbers. The patient reported by Cohen et al. (1994) suffered from deep dyslexia and could read half of the presented words (12/24), but no non-words (1/24). When reading multidigit numbers, he was also better at reading familiar, encyclopedic numbers (e.g., 1789, French Revolution) than unknown numbers (24/52 vs. 10/52). Similarly, Delazer and Girelli (1997) describe an aphasic patient who performed better in reading Arabic numbers in a semantic context (e.g., he could read “164” when preceded by “Alpha Romeo”) than without context or a non-associated context. The opposite pattern has also been described in the case-study by Cappelletti et al. (2008). In this case, the patient suffered from semantic dementia and was impaired in reading encyclopedic numbers, thus numbers corresponding to facts retrieved from stored knowledge. While still being able to process non-encyclopedic numbers (3 x 3 = 9), he was also unable to learn new encyclopedic facts when they included numbers. The patient’s impairment was presumably due to representational problems concerning the visual Arabic input levels, and not from retrieval of verbal forms, as he did not benefit from priming or from multiple choice situations.

More recently, an fMRI study complemented these neuropsychological dissociations by revealing different brain activations for numbers when processed as quantities vs. dates of famous historical events (Gullick & Temple, 2011). In the study, the authors used both unknown numbers and known famous dates. Before the experiment, they verified that participants knew the dates and allowed them a short study session to refresh memory traces. Then, they proposed two different tasks. Either, participants had to process numbers as quantities in a numerical comparison task where a pair of numbers was presented and participants had to choose which is larger/smaller, either they had to process them as events and rank the latest/earliest event of the pair. In the first task, numbers activate only the intraparietal sulcus bilaterally, as expected from the known regions involved in quantity processing (Ansari, 2007; Nieder, 2004). In the second task, numbers activate not only bilateral IPS but also extra regions related to semantic processing and fact retrieval: the temporal pole, and superior frontal gyrus.

Other studies in healthy individuals on this issue are scarce. Using a priming design to investigate the existence of stored representations for encyclopedic numbers, Alameda et al. (2003) required participants to name multidigit numbers preceded by related or unrelated masked or unmasked primes. So, as an example, participants had to read aloud the Arabic number “747” and it was preceded by “Boeing” (related prime), or by “Porsche” (unrelated prime). Participants were faster with the related prime, which seems to suggest that there are representations for familiar numbers, that could be activated by associated verbal labels. However, the effect could also stem from pre-assembled expressions in phonological format (e.g., Noël & Seron, 1995), as the task required participants to read aloud the numbers. Therefore, in the second experiment, the authors designed a “number decision task” not requiring any verbal output: participants had to answer with keypress whether a presented item was a legal/illegal number. Legal numbers were made of digit strings, while “non-numbers” were a mixture of digits and letters. Again, numbers were preceded by related or unrelated verbal primes. In that case, facilitation effects strongly suggest the existence of stored representations for known familiar numbers, automatically activated by associated semantic (non-numerical) knowledge, thus entailing the existence of an input lexicon for Arabic numbers.

Our study goes a step further in assessing the existence of a mental store, or lexicon, for numbers, by using a design that does not involve any semantic or verbal associate, no priming and no explicit processing of the whole multi-digit number. Indeed, to test if some numbers are represented in an input Arabic lexicon as lexical items, we decided to compare known dates to unknown numbers, and to test experts with dates, thus historians. This choice was made in order to design a homogeneous set of stimuli, supposedly known by all participants. We used a paradigm from the reading literature that has revealed a word-superiority effect (WSE, Reicher, 1969; Wheeler, 1970). The WSE initially showed faster RT and better accuracy to identify a letter in a two-alternative-forced-choice (2AFC) task when it belongs to a word than when the letter is presented in isolation, suggesting an advantage due to lexical activation. This paradigm has been replicated numerous times in the literature, and it was also shown that a letter is better identified when belonging to a word than a pseudoword (PW) (Grainger et al., 2003; Grossi et al., 2009). The WSE has been shown to correlate with behavioral lexical decision performances (Hildebrandt et al., 1995) and to emerge with the building of an orthographic lexicon (Chase & Tallal, 1990; Coch et al., 2012; Grainger et al., 2003). It has generally been interpreted as reflecting top-down influences from lexical levels of representation on letter-identification processes in the general framework of the interactive activation model (McClelland & Rumelhart, 1981; Rumelhart & McClelland, 1982), or in cascaded models of visual word processing (Coltheart et al., 2001).

Following the logic of this paradigm, here a number that could be a known date or not was briefly presented, without participants being aware of this manipulation. It was followed by a mask and an alternative choice between a target digit and another digit (2AFC task), placed above and below the corresponding position in the string (Figure 1).

Our general hypothesis was that in contrast to unknown numbers, dates are stored in an input lexicon and thus they should be sensitive to similar manipulations than words by comparison to non-words. More specifically, we hypothesized better digit identification performance in the case of dates than in the case of unknown numbers.

Method

Results

Discussion

In the current study, we investigated the possible existence of an Arabic input lexicon for encyclopedic numbers, automatically accessed on the basis of the input form, in healthy individuals. Evidence for such an encyclopedic number lexicon was until now lacking, as previous findings could be explained by explicit semantic elaboration and processing (Gullick & Temple, 2011), or by activation of semantic and verbal associates (Alameda et al., 2003). Here, testing historians with high knowledge of dates, we compared the processing of dates to unknown numbers with the Reicher-Wheeler paradigm borrowed from the reading literature (word superiority effect, WSE, Reicher, 1969; Wheeler, 1970).

In this experiment, participants were not informed that there were two categories of numbers, dates and non-dates, or two categories of letter-strings. They had to identify which of two characters had been presented in the preceding stimulus. Overall, the results show a benefit for known items on character identification. On reaction times, participants were faster for known items, independently of these items being words or dates (no interaction with context). On accuracy, performance was better for known than unknown items, but this was modulated by position differently for words or dates. The same was found on IES: the benefit for known over unknown items was mainly occurring at position 2 for numbers, and at position 3 for words.

Before discussing the potential reasons for these position differences, we would like to highlight our main novel finding: dates are processed differently than non-dates, similarly as words by comparison to non-words, in a totally implicit task where participants were not required to judge items as dates, or to activate any semantic knowledge about dates. Since the WSE is classically interpreted as reflecting the impact of stored orthographic representations in the context of word processing (Grainger et al., 2003), the results of this experiment confirm that dates may be considered as stored items in an Arabic input lexicon. This idea was first put forward in neuropsychological studies, like the patient described by Cohen et al. (1994) who was better at reading familiar Arabic numbers (e.g. the dates 1789, 1918) than unfamiliar ones. Also, even for familiar numerals that he could not read, he made comments that indicated he had access to the (non-numerical) meaning of the item. The authors, by analogy with word reading, proposed the existence of a “surface” reading route and a “deep” (i.e., nominal semantic) reading route, that would involve an input lexicon for encyclopedic Arabic numbers. Other case-studies showed a similar dissociation, with better performance for encyclopedic than unknown numbers (Cipolotti, 1995; Delazer & Girelli, 1997), as well as the reverse (Cappelletti et al., 2008).

Besides patients’ studies, previous data on healthy individuals suggested different memory retrieval processes for personally familiar numbers (Dickson & Federmeier, 2018), distinct brain regions activated by famous dates when processed as events rather than as quantities (Gullick & Temple, 2011), as well as the existence of an input lexicon for encyclopedic numbers (Alameda et al., 2003). While the two former studies relied on explicit activation of number’s semantics, either personal (Dickson & Federmeier, 2018) or encyclopedic (Gullick & Temple, 2011), the later relied on implicit priming effects (Alameda et al., 2003). Interestingly, facilitation effects due to related primes were shown on multidigit Arabic numbers, both in reading aloud and in lexical decision tasks. However, the use of a verbally-related prime could not preclude that the priming benefits were due to the activation of a pre-assembled expression in phonological format, or to links that could exist in an orthographic input lexicon for words associated with the Arabic number. Here, this potential explanation does not hold, because we did not use any verbal primes or labels and no information or context. Thus, contrarily to previous studies on healthy individuals (Alameda et al., 2003; Gullick & Temple, 2011), we show that dates have triggered automatic activation of stored number forms. Our results clearly show that the link between known Arabic numbers and encyclopedic knowledge does not systematically require extensive and slow top-down elaboration of the stimulus, and does not require activation of an associated verbal label (either explicit, e.g. the patient of Delazer & Girelli, 1997, or implicit, e.g. Alameda et al., 2003). Our experiment shows that dates trigger a faster identification of constituent digits, like words speed up processing of constituent letters, and on the contrary to unknown items. This can be understood in an interactive-activation or cascaded model of reading, where items that possess an entry in the lexicon feedback activation to the level of digit identification (McClelland & Rumelhart, 1981). ERP studies showed indeed that lexical factors may facilitate letter string processing between 160ms and 200ms, such as frequency (Assadollahi & Pulvermüller, 2003; Hauk & Pulvermüller, 2004; Sereno et al., 1998), or lexicality (Martin et al., 2006).

In a general cognitive architecture of number reading, some authors have debated over the existence of a several routes for reading numbers (overview in Brysbaert, 2018): a purely asemantic route (Dehaene, 1992, 2011; Seron & Deloche, 1984), a semantic route activating cardinal/ordinal value of numbers (Brysbaert, 1995; McCloskey, 1992), or a lexical route for encyclopedic numbers (Cipolotti, 1995; Cohen et al., 1994).

Discrepancies in findings and thus, in theoretical proposals, might be reconciled if considering the co-existence of these multiple routes for reading numbers, triggered both by the task and the nature of the stimuli. Indeed, one proposal stated that Arabic numbers are resembling pictorial images more than words (Fias et al., 2001) therefore accessing semantics (magnitude) in a direct and obligatory manner. Evidence in favor of this idea initially came from STROOP-like paradigms, on 1-digit Arabic numbers. When participants had to name number words, they were not affected by the (in)congruent presence of a digit; but when they had to name an Arabic number they were affected by the presence of a number word. This suggested that words could be read totally asemantically, using the sublexical conversion route (letters-sounds), while digits could not, and automatically activate their magnitude representation. This might be true indeed for reading 1-digit Arabic numbers. Since then, however, other experiments also provided evidence that 2-digits numbers are not processed holistically as a picture, but are decomposed (Nuerk et al., 2001) and also that they can be read by a direct route relating Arabic input and verbal output forms (Herrera & Macizo, 2012; Ratinckx et al., 2005). Finally, familiar numbers trigger another route than unknown numbers: this was suggested by patient studies and we add here evidence that it might be the case in healthy individuals, even in an implicit task. In this lexical route, the coding of internal elements obeys the same law as letters in words, thus digits identity and position within the chain are processed in parallel like letters that constitute a word.

As concerns the position effects reported here, serial position effects are widely described in the literature on letter identification, where several constraints play a role in explaining better or worse performance for identifying/reporting letters depending on their position in the sequence. To examine position effects, many studies used 5-elements strings: the classical finding is a W-shape in accuracy and/or a M-shape on reaction times, both reflecting better performance on the central and the external letters, and worse performance at the second and fourth positions. This typical variation of performance according to position reflects the combination of several factors. First, the structural properties of our visual system induce better identification at the fovea (thus, the central position), and a decrease of acuity with eccentricity towards external positions (for instance, Nazir et al., 2004). Second, external positions benefit from a lack of lateral crowding as they are flanked by only one other character and not two (inter-letter interference; Bouma, 1970, 1973). Finally, letter identification is better on the left (especially the first letter) than on the right of central fixation in languages read from left to right, which would reflect an attentional bias (Aschenbrenner et al., 2017; Nazir et al., 2004; Pitchford et al., 2008) or an adaptative asymmetry in the receptive fields for retinotopic letter detectors, favoring the leftmost information and resulting from an optimization for processing initial letters in languages read from left to right (Chanceaux & Grainger, 2012; Grainger & Van Heuven, 2003; Tydgat & Grainger, 2009). There is a debate in the literature regarding the origins of these effects and also, their specificity to letters or stimuli typically processed in strings (letters and digits; Mason, 1982; Tydgat & Grainger, 2009), or to all types of visual objects (Gomez et al., 2008). Here, although we did not test external positions, we do observe overall an advantage of position 2 over position 3, both for numbers and letters (faster RT, better accuracy). This is expected in the general pattern described above, given that the center fixation was located between position 2 and 3: it shows a slight advantage on the left rather than the right of the fovea. What differs between numbers and letters is the location of the benefit due to the lexical status (known vs. unknown): for letters it is significant only at position 3, while for numbers it is the case only at position 2. On words, the WSE (better performance in identifying letters in words than non-words) at position 3 only was reported in another study using 4-elements stimuli with a CVCV structure, although in a shallow orthography (Italian; Ripamonti et al., 2018). On numbers, we can only speculate on a few reasons why they might have driven a specific pattern at position 2. First, because we used 4-digit dates, most of them started with the digit “1”, therefore the first position was not very informative. Thus, the second position was the first character of the string to be informative/ to potentially activate items as distinct from one another in memory, which could induce better performance, similarly as the first letter-advantage in words. In this perspective, the finding at position 2 would only reflect a list-context-effect, and we should not observe similar findings if the first digit would vary across items (which is not possible with 4-digit dates but would be possible using 3-digit dates). It would also mean that a similar advantage at position 2 could be found in words if using a list of items all starting with the same letter. A second potential explanation is linked to the specific processes deployed to read Arabic numbers. As stated in the introduction, multidigit numbers are parsed in triplets from the rightmost position, therefore, the second digit of a 4-character string was also the first digit of the triplet, and although speculative, the first element of a triplet might have a specific status. However, this would mean that both the lexical recognition process and the transcoding process would be triggered in parallel. Third, several dates included the digit 9 in second position (8/18). Among the 6 items that were retained for analysis (see Methods), only two included the alternative choice on that position. To exclude the possibility that these two items could have induced a generally faster RT at position 2 (because the digit 9 would be expected at that position), we checked RT for numbers containing 9 at position 2. They were actually slower (1117ms) than the RT for all others items not containing 9 at this position (1057ms); thus if anything these results go against the idea of faster RT at position 2 for dates because of the repeated presence of digit 9.

To sum up, in this study we found a significant effect of the status of numbers on character identification, with a “date superiority effect”, i.e., an increased performance (faster reaction times) for recognizing digits belonging to dates by comparison to non-dates. This effect was however present only at position 2 in the string, and several interpretations of the position effect have been put forward. There are some limitations to our study. First, we used a relatively small sample of participants (N = 24) and items (N = 18), due to the specific domain of knowledge that was tested (History). This limits the power of our results (Brysbaert, 2019). Second, although competitors for dates were also known dates (see Appendix), we could not match them in frequency (like words and competitors) as such frequency lists do not exist, and we did not test their knowledge in the matching task. Thus, it may be that some items were less familiar or frequent than others.

Further studies should investigate how our findings can be generalized to other types of encyclopedic numbers besides dates, like labels (Boeing 747), codes or phone numbers. Here we tested experts with dates, for a matter of experimental homogeneity, but in everyday life we all have personal number facts that are presumably stored in such a lexicon, and they may vary in length, chunking structure and type of verbal counterparts. For instance, stock managers label products by strings of 3 digits where each digit has a specific meaning, while pin codes or phone numbers are chunked in strings of 2 or 3 digits. Further research should aim at understanding how such a lexicon is organized: what defines neighborhood between Arabic numbers? Could we evidence similar principles of competition, activation/inhibition as in word processing? How stable are these representations? How are they influenced by frequency and length? Are both reading routes (lexical and non-lexical) triggered in parallel? Is the magnitude information nevertheless activated for these numbers? These questions show that (re)opening this field of research goes beyond the simple conclusion that Arabic numbers might be stored in an Arabic input lexicon, they point to the broader issue of generalizability of the principles that have been discovered in the domain of word reading and the orthographic lexicon.

Taken together, our results show the existence of a lexical route for reading encyclopedic numbers in healthy participants. Activation of this Arabic input lexicon can be observed without using any verbal priming and it occurs during implicit processes (i.e. not entailing any decision-taking), indicating that it does not require extensive top-down elaboration.