Imagine living in a dark cave, with your entire understanding of the world based on shadows on the wall. Sounds unrealistic and terrifying, right? However, this allegory presented by Plato is an apt metaphor for our brain’s perception of the real world. While we might believe we perceive reality in its entirety, our brain can only provide a shadowy representation of the variables in our environment, and our decisions are based on these shadowy representations.
A comforting thought might be that numbers are a universal language for Western countries and, specifically, for those who use Arabic numerals. For instance, when the price of a good is marked as $14, it conveys an unambiguous and specific value, meaning that one would unequivocally expect to pay exactly that amount. However, experiments show that people make mistakes when dealing with Arabic numerals. For instance, under time constraints, the closer two numbers are in value, the more challenging it becomes for us to rapidly and accurately pinpoint the larger one. These mistakes are very similar to those made in psychophysics tasks involving physical stimuli, such as comparing the length of segments or averaging the orientation of tilted lines. These results, together with neurobiological studies, suggest the existence of a representation system for numbers that is similar to how we interpret physical stimuli.
A popular idea in theoretical neuroscience is that while the brain’s computational abilities have inherent limits, leading to imprecise representations, these representations are optimal within those constraints. This theory, called “efficient coding”, suggests that our brain’s perceptions are influenced by how often these magnitudes are encountered (i.e., the prior). For example, vertical and horizontal orientations are perceived with more clarity than oblique ones, likely because they’re more common in our environment.
A recent study (https://www.nature.com/articles/s41562-022-01352-4), led by Columbia postdoc Arthur Prat-Carrabin and published in Nature Human Behaviour, delves into whether our brain treats numbers the same way it treats physical stimuli. In their experiment, participants were asked to determine which series of numbers, red or green, had a higher average value. For instance, as shown in Figure 1, the number 79.60 would flash in red on the computer screen, followed by 44.92 in green, and so forth. Participants were tasked with rapidly and intuitively calculating the average of the red and green numbers to determine which sequence had the greater average. To investigate how the frequency at which numbers were encountered impacts their representation, numbers were drawn from different distributions: one, in which smaller numbers were more frequent (Downward prior); another, in which all numbers had equal chances (Uniform prior); and a third, in which larger numbers were more frequent (Upward prior).
To analyze the participants’ decisions, the researchers compared their answers with multiple computational models characterized by two components: first, whether or not the numbers are encoded with a bias that depends on the value of the number, and second, whether or not the imprecision (the noise) with which numbers are represented varies with the number. Their results showed that the model that best aligned with the participants’ answers had to include both components. Notably, less common numbers are fuzzier in their perception. So, when using the Downward prior, bigger numbers are encoded with more noise, while with the Upward prior, the smaller numbers are the noisiest ones.
This discovery not only supports the “efficient coding” theory, which posits that the brain encodes and represents information in the most efficient way possible, but also showcases its applicability beyond just physical perceptions. Whether we’re assessing the speed of a car, the talent of a dancer, or the sweetness of a cake, our brain might use a universal mechanism to represent these variables. This mechanism dynamically adjusts to the statistical distribution of numbers that are expected or experienced, which implies that our understanding of numbers and magnitudes isn’t static but can be influenced by our prior experiences and expectations. In the near future, we might be able to design environments that help people refine their perceptions (such as by crafting digital games to enhance the consumer’s responsiveness to certain prices), allowing them to better discern specific value ranges and improve their decision-making.