This study had a twofold purpose. The first was to assess whether non-symbolic estimation of time, space, and number is processed by the same cognitive mechanism. We analyzed the accuracy of participants in temporal, spatial, and numerical discrimination tasks with the assumption that, as predicted by ATOM, high ability in one domain (i.e., numerical) should correlate with high ability in the other two domains (temporal and spatial). Second, we wanted to assess whether non-symbolic numerical abilities serve as a foundation for symbolic numerical abilities. We correlated the performance of participants who were required to quickly estimate which groups of dots were more numerous with that reported when participants were required to perform mental calculation and mathematical reasoning tasks.
With respect to the first purpose, we found that the ability to discriminate temporal, spatial, and numerical dimensions was strongly affected by the ratio, in agreement with Weber’s law. This aligns with a large body of experimental evidence accumulated in cognitive [1, 41], developmental , and comparative [43, 44] psychology. As predicted in the literature [12, 13], the only exception was found in the performance of the numerical task within the subitizing range: The typical signature of the OTS—ratio insensitivity—was found in the range of 1–4. However, we found a pattern of data that contradicted ATOM. Indeed, no correlation was found among the three tasks: Some participants were more accurate in temporal and some in spatial or numerical tasks, but there was no evidence that more accurate participants in one domain also performed better in the other domains.
It is interesting to note that participants exhibited an overall worse performance (lower accuracy, higher reaction time and higher Weber fraction) in the temporal discrimination task. As such, it is worth remembering that the temporal task was an auditory task, and we cannot exclude the possibility that the different sensory modality might have played a key role. There is indeed an open debate as to whether time estimation varies as a function of the sensory modality involved: While some studies have reported inter-sensory difference in time estimation [45, 46], others have shown no difference in accuracy between visual and auditory stimuli [47, 48]. This debate extends far beyond the scope of this study. However, even assuming that time estimation might be somehow different in the auditory modality, participants’ performance in temporal, spatial, and numerical tasks still would have been correlated according to ATOM, a condition that did not occur here.
In line with our results, DeWind and Brannon  found evidence of a dissociation between number and space processing. The authors administered a simple numerical training, finding improved ability in the numerical but not in the spatial task. This is incompatible with number and space being processed by the same mechanism. The authors suggested a weaker version of ATOM, according to which space, time, and number would be at least partially differentiated. After all, the fact that not all aspects of time, space, and numbers may have a common origin was also advanced by Walsh  in the same paper in which ATOM was theorized for the first time. Some tasks are expected to be entirely solved using temporal/spatial/numerical systems outside of the common mechanism. Our results concur with this interpretation of ATOM.
With regard to weaker interpretations of ATOM, it has been also suggested that similar or even identical accumulators may work simultaneously; for instance, numerosity and duration could be processed independently by two different accumulators before converging in a common system [24, 49]. Alternatively, there may be separate stimulus-processing pathways, especially for number-space processing. There is indeed evidence that space and number may share a common representation (a mental number line) and influence each other in tasks where they are processed together . A positive correlation between spatial and numerical tasks was found by Thompson and Siegler  and Booth and Siegler . In this study, we did not find a correlation between number and space. It is possible that the correlation among non-symbolic magnitude estimations may be affected by the type of task and/or the characteristics of the population. For instance, in the above-mentioned studies [51, 52], the authors tested children (5–8 years old) and used production tasks (such as drawing a line that is x long, creating a jar that has x candies in it). On the other hand, we tested adult humans for judgments of relative magnitudes (which is the longer line or the more numerous group). Some numerical abilities might be more closely related to spatial abilities, as well as the relation among time, space, and number might change over the course of cognitive development. A recent study found evidence of a single magnitude system in infants , while the debate is still largely open with respect to adults [15, 23, 24, 54]. To date, there is no evidence that the supposed common magnitude system would work similarly from childhood to adulthood. Longitudinal and/or cross-sectional studies investigating temporal, spatial and numerical abilities at different ages are needed in order to test such a hypothesis.
Regarding the second purpose of this study, we found a positive correlation between non-symbolic (within and outside the subitizing range) and symbolic numerical abilities (both mental calculation and mathematical reasoning). Previous studies found a similar correlation [1, 22, 29]. These results are believed to suggest that non-symbolic numerical abilities might serve as a building block upon which symbolic numerical abilities are based. In this sense, the precision of non-symbolic numerical abilities would facilitate/affect comprehension of arithmetic and mathematics. However, as stated by Butterworth , correlations are not indicative of causation, and it might be possible that poor performance in non-symbolic numerical tasks is the consequence of poor mathematical ability (instead of the cause). We can only speculate on this point. However, the latter hypothesis appears to be less likely in our view. Non-symbolic numerical abilities are known to be present at birth  and are based on a core number system that improves in precision well before the acquisition of symbolic language [56, 57]. It was found that the ANS precision of 3- to 4-year-old children—before they have begun formal mathematics instruction—predicts their mathematics scores at age 5 or 6 years. In contrast, it does not predict their scores on other cognitive tasks, such as vocabulary size or the ability to identify colors or letters . Above all, Castronovo and Göbel  found that experts in mathematics do not exhibit better performance in non-symbolic numerical abilities, thus excluding the possibility that long-term training in mathematics could easily shape non-symbolic numerical abilities. Further studies are needed on this issue. In the absence of more adequate explanatory frameworks, the positive correlation between non-symbolic and symbolic numerical abilities allows for the possibility that non-symbolic numerical systems can scaffold symbolic numerical systems.
It would now be a challenge to understand why some studies found significant correlations between symbolic and non-symbolic numerical tasks [1, 29], whereas other studies have not [32, 36]. It was recently suggested  that detecting a positive correlation between non-symbolic and symbolic numerical abilities may depend on sample size and/or other characteristics of the population. For instance, different sub-types of dyscalculia has been hypothesized , as not all studies support the idea that dyscalculia may be caused by impairment of the non-symbolic numerical systems. Developmental differences in the ANS need to be taken into account, too: Given that non-symbolic numerical abilities are known to increase in precision over the course of cognitive development [1, 42, 56], differences in age among the studies' subjects may explain the inconsistencies reported in the literature. Also, the different results may be ascribed to the different acuity metrics for measuring non-symbolic numerical abilities : accuracy vs. reaction time, numerical ratio effect vs. internal Weber fraction, etc. Here we analyzed the three main variables adopted in literature to study the correlation between symbolic and non-symbolic numerical abilities (accuracy , reaction time  and internal Weber fraction ), but we cannot exclude that other composite measures may capture a different variation of performance .
Last but not least, the nature of the tasks might play a key role. For instance, it is possible that some symbolic numerical abilities are more influenced by non-symbolic numerical systems than others. As far as we know, no study has used the symbolic numerical task adopted in this work: Mental calculations were chosen by the experimenters and no other studies use the sub-scale of mathematical reasoning of the WAIS-R. If the emergence of a positive correlation is context-dependent, the specific items selected here might explain our results. Future studies are required to find out under what circumstances the tasks are related and under what circumstances they show little or no relation to one another. Part of the problem could be tackled by presenting participants with both symbolic numerical tasks which are known not to correlate with non-symbolic numerical tasks (e.g., calculation subtest of the Woodcock–Johnson III tests of achievement, see ) and the symbolic task adopted in this study. This would provide a finer comparison with the existing literature, enabling us to compare the performance in different symbolic tasks within the same population.
It is worth noting that performance in the symbolic numerical task did not correlate with performance in non-symbolic estimation of space and time. This is again incompatible with a strong version of ATOM. Indeed, another potential prediction of ATOM is that if less/more accurate non-symbolic numerical skills underlie lower/higher mathematical abilities, then less/more accurate skills are also expected in temporal and spatial discrimination, assuming the same magnitude system.
We also found that symbolic numerical abilities positively correlated with the performance exhibited in numerical estimation both within and outside the subitizing range. Previous studies suggested the importance of the ANS in acquisition of symbolic numerical abilities [61, 62], while others remarked on the role of the OTS . Still others considered the combination of the two cognitive systems to be crucial [64, 65]. Our results align with the latter hypothesis, suggesting that both the OTS and the ANS are involved in the acquisition of formal mathematics.
We cannot exclude the possibility that an increased sample size might have provided us with a clearer picture. However, we feel this is unlikely, as no marginally significant results were found. As such, it is worth noting that Holloway and Ansari  tested 87 participants (more than twice the number of participants tested here) without finding any correlation between non-symbolic and symbolic numerical abilities; in this sense, it is unlikely that sample size alone can explain the presence/absence of correlations here reported. Furthermore, the correlation among reaction times mirrored the correlations observed for accuracy, thus reinforcing our conclusions. The only difference between the two dependent variables was the positive correlation (in reaction time) between non-symbolic numerical estimation in the OTS and in the ANS range, which does not change our main conclusions. Also the internal Weber fraction of non-symbolic numerical estimation was significantly correlated with mental calculation and mathematical reasoning. To this purpose it is worth noting that the average Weber fraction of 0.16 in the current study is concordant with previous values reported in literature (e.g., 0.17 ), which further aligns this work with the existing literature on non-symbolic numerical abilities.
As a last note, one may argue that the positive correlation observed between non-symbolic and symbolic numerical abilities might have been due to concurrent factors, such as different motivations, attention levels, and/or working memory. If so, we should have observed positive correlations in all tasks. Above all, the performance in the control test—which involved no magnitude processing—did not correlate to any task, which seems to exclude the possibility that factors not related to magnitude processing might explain our results.