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Defining “What Is A Number”: Topology-defined Units in Numerosity Perception

Author: Update time: 2015-09-29
What is a number? The answer to the age-old and fundamental question of philosophy has increasingly benefited from recent psychological and neuro scientific investigations. A recent article entitled Topology-defined Units In Numerosity Perception in Proceedings of the National Academy of Sciences of the USA (by ZHOU Ke and CHEN Lin as corresponding authors from Institute of Biophysics, Chinese Academy of Sciences), which proposed that numerosity may be definable in terms of topological invariants, such as connectivity (the number of connected components) and the inside/outside relationship (surrounded/surrounding components). In other words, the primitive units counted in numerosity perception may be units based on topology.

The authors believe that if the primitive units to be counted are essentially defined by topology, people can predict some experimental phenomena that are not necessarily consistent with their intuition about numerosity perception, but with topology. For instance, intuitively, it seems that the inside/outside relationship does not exert fundamental effect on numerosity. Nevertheless, the topological analysis predicts that enclosing dots, like connecting dots, may lead to numerosity underestimation, because multiple dots enclosed within a hollow figure should be perceived as a holistic perceptual unit.

To verify the nature of topological invariance of numerosity, the authors manipulated the numbers of items connected or enclosed in arbitrary and irregular forms, while controlling low-level features (e.g., orientation, color, and texture density). They also use subjects which perform discrimination, estimation, equality-judgment, as well as a wide range of presentation-durations and test small and large numbers. Besides, neural tuning curves to numerosity in the intraparietal sulcus were obtained by using fMRI-adaptation. Results are consistent with the topological account: connecting or enclosing items will lead to robust numerosity underestimation, and the extent of underestimation will increase monotonically with the number of connected or enclosed items; and neural tuning curves to numerosity demonstrate that numbers represented in the intraparietal sulcus are largely determined by topology.

The topological approach has contributed to the study of the fundamental philosophical question--- what is a number, by means of psychology and neuroscience in a precise and concrete way as shown in this series of behavioral as well as fMRI experiments. The results lead to the intriguing suggestion that numerosity, as a basic invariant property of the environment, may be formally described in terms of topological invariants.

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