Core systems of number

post by Bruce W. Lee (bruce-lee) · 2024-02-09T02:19:03.207Z · LW · GW · 0 comments

This is a link post for https://www.sciencedirect.com/science/article/abs/pii/S1364661304001317

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The exploration of numerical cognition reveals foundational elements present from infancy, shared by various animal species, highlighting an inherent numerical understanding that does not require formal education or cultural influence. Feigenson, Dehaene, and Spelke (2004) discuss two core systems underpinning our ability to comprehend and manipulate numbers: one for approximating large numerical magnitudes and another for precisely representing small numbers of objects​.

The first core system, evident in both infants and animals, allows for the discrimination of large numerical quantities through an approximate sense but is constrained by a ratio limit. This limitation manifests as an inability to precisely distinguish between closely numbered sets beyond a certain ratio, a trait observable across species including humans, monkeys, and rats, underscoring a shared, abstract representation of numerical magnitude​​​​.

In contrast, the second core system enables the exact representation of small numbers, up to three or four, beyond which precision falters. This system's uniqueness lies in its exactitude for small quantities, a feature that does not scale with larger numbers due to inherent limitations in cognitive processing. Remarkably, this precision in representing small numbers is also found in non-human animals, suggesting a common evolutionary heritage​​​.

These core systems not only elucidate our intuitive grasp of numbers but also lay the groundwork for understanding the complexities of mathematical concepts developed later in life. They represent a universal aspect of cognition, foundational to the numerical capabilities observed across different species and developmental stages​.

(This excerpt is worthy of just copy and pasting from the paper.)

Why is number so easy and yet so hard? Although studies of human infants have not definitively answered this question (see Box 2), they offer several suggestions. First, number is easy because it is supported by core systems of representation with long ontogenetic histories. One system serves to represent approximate numerical magnitudes independently of non-numerical quantities. Because this system is active early in infancy, humans are attuned to the cardinal values of arrays from the beginning of life. The other system serves to represent numerically distinct individuals of various types and allows multiple computations over these representations. These computations include forming summary representations of the individuals’ continuous properties, and representing the number of individuals in an array. Because this second system is also active in infancy, concepts of ‘enumerable individual’ and ‘adding one’ are accessible throughout our lifetimes. Numerical reasoning might be easy, and numerical intuitions transparent, when they rest on one of these systems. 

Second, number is hard when it goes beyond the limits of these systems. When one attempts to represent an exact, large cardinal value, one must engage in a process of verbal counting and symbolic representation that children take many years to learn [60,61], that adults in different cultures perform in different ways [62], and that people in some remote cultures lack altogether (Gordon, unpublished). When humans push number representations further to embrace fractions, square roots, negative numbers and complex numbers, they move even further from the intuitive sense of number provided by the core systems. 

What drives humans beyond the limits of the core systems? If the human mind were endowed only with a single system of core knowledge, then humans might never venture beyond its bounds. We are endowed, however, with two core systems of numerical knowledge and with other systems for reasoning about physical, living and intentional beings (e.g. [63–65]). As we apply these different systems to the same objects, events and scenes, we appear to be driven to reconcile the representations that they yield. Three-year-old children might show this drive when they struggle with their representations of approximate magnitudes and of numerically distinct individuals so as to learn the meanings of words like ‘seven’, a concept whose meaning is guaranteed by neither core system. Newton and Leibniz may have shown a similar impulse when they independently invented the calculus, stretching their systems of numerical and mechanical knowledge so as to reconcile them. Nothing guarantees, however, that the intuitions provided by distinct core systems can be reconciled into a single system of consistent, transparent and accessible truths. Despite the great advances in human physical and mathematical concepts over cultural and intellectual history, this consilience continues to elude us.

Furthermore, neuroscientific research supports the distinct neural correlates associated with these core systems. The system for approximate numerical magnitude is linked to the bilateral horizontal segment of the intraparietal sulcus, a finding consistent across neuroimaging studies in humans and electrophysiological recordings in monkeys. These studies reveal neurons tuned to numerical quantities, with precision diminishing as numbers increase, highlighting the inherent imprecision of the approximate number system​​​.

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