Abstract: Clarifying the relationship among mechanisms underlying nonsymbolic numerical cognition is one of the most critical questions in the numerical cognition field. From a novel perspective, the hypothesis that three-number processing systems exists provides a plausible explanation for this relationship. According to this hypothesis, distinct mechanisms are involved in the number processing of nonsymbolic stimuli within different number ranges. Small numbers (1~4) can be appraised rapidly and errorlessly based on the activation of subitizing system. Moderate numbers are proposed to be processed spontaneously with a constant error rate of about 20% due to the activity of numerosity system. Typically, Weber's law of number perception is demonstrated in this number range. For large numbers, the stimulus number relationship is suggested to be inferred indirectly via density analysis, and number processing, which is fast and has an error rate proportional to the square root of the stimulus number, is mediated by the activity of density system.
A series of studies have revealed different behavioral features, as well as distinct Event-related potentials (ERP) features, among number tasks based on these three systems. For subitizing and density systems, better processing efficiency is demonstrated by a shorter reaction time and a lower error rate, whereas a higher dependence on attention resources is also a characteristic of these two mechanisms. Neither subitizing nor density mechanisms are correlated with math ability for children in school. For the numerosity mechanism, on the contrary, lower efficiency and independence of attention are shown, and the accuracy of numerosity comparisons is suggested to be significantly correlated with math scores for school children. ERP features are also suggested to be distinguishable for these three mechanisms. An early component related to attention is typically found under a subitizing mechanism rather than under numerosity or density mechanisms. P2p showed a distance effect, namely, higher P2p amplitude correlates with more errors in comparison tasks, as the ratio of the numbers to be compared approaches “1”, under both numerosity and density mechanisms. In general, the amplitude of P2p (about 200 ms after the onset of stimuli) is larger for the density mechanism compared with that for the numerosity mechanism, whereas the P2p component is more sensitive to the change in number within the numerosity range than within the density range.
There are two major challenges for the three-number processing hypotheses. First, as proposed by some researchers, even when the numerosity and density mechanisms are distinguished by different modes of Weber fractions, more direct evidence is still needed to demonstrate that the processing bases of these two mechanisms are numerosity and density, respectively. Second, the narrow range described in the original hypothesis may induce discontinuity between the three-number processing hypotheses and the classical hypothesis, for example, the approximate number system hypothesis or ANS hypothesis. To solve these problems, it is proposed that the cause for switching from numerosity to density should be reanalyzed and that the hypothesis that the numerosity mechanism is activated in a wider number range should be considered.
Some experimental evidence indicates parallel activation between subitizing and numerosity systems as well as between density and numerosity systems. The parallel activation hypothesis is also noteworthy. Multiple number processing systems could be activated simultaneously, and cognition relies on the system whose processing result is more precise to achieve the best processing results.

Key words: subitizing system, numerosity system, density system, parallel activation