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

Sep 28, 2015

What is a number? The answer to this age-old and fundamental question of philosophy has benefited from recent psychological and neuro scientific investigations. ZHOU Ke and CHEN Lin from Institute of Biophysics of Chinese Academy of Sciences 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. This article entitled "Topology-defined Units In Numerosity Perception" was published in PNAS.

The researchers believed 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. However, the topological analysis predicts that enclosing dots, like connecting dots, may lead to numerosity underestimation as 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 researchers 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 used subjects which perform discrimination, estimation, equality-judgment, as well as a wide range of presentation-durations and tested small and large numbers. Besides, neural tuning curves to numerosity in the intraparietal sulcus were obtained by using fMRI-adaptation.

The results are consistent with the topological account. Connecting or enclosing items leads to robust numerosity underestimation, and the extent of underestimation increases monotonically with the number of connected or enclosed items. Neural tuning curves to numerosity demonstrate that numbers represented in the intraparietal sulcus are largely determined by topology.

The topological approach contributes 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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