The Psychological Scaffolding of Arithmetic

Grice, Matt; Kemp, Simon; Nj, Morton; Grace, Randolph C.

doi:10.31234/osf.io/myq4n

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…The boundaries between convex and concave surfaces are important for identifying components of objects (Koenderink & Van Doorn, 1982;Biederman, 1987). There is an overall bias for perceiving objects as convex, as in 'hollow-face ' and 'hollow-potato' illusions (Dobias, Papathomas, & Vlajnic, 2016;Gregory, 1970;Hill & Bruce, 1994;Hill & Johnston, 2007;Langer & Bulthoff, 2001), which appears even in 6-month old infants (Corrow, Mathison, Granrud, & Yonas, 2014). A bias for convexity has also been shown with 3D 'polycube' stimuli (consisting of random configurations of 10 individual cubes),…”

Section: Convexitymentioning

confidence: 99%

The psychological scaffolding of arithmetic.

Grice¹,

Kemp²,

Morton³

et al. 2024

Psychological Review

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Where does arithmetic come from, and why are addition and multiplication its fundamental operations? Although we know that arithmetic is true, no explanation that meets standards of scientific rigor is available from philosophy, mathematical logic, or the cognitive sciences. We propose a new approach based on the assumption that arithmetic has a biological origin: Many examples of adaptive behavior such as spatial navigation suggest that organisms can perform arithmetic-like operations on represented magnitudes. If so, these operations—nonsymbolic precursors of addition and multiplication—might be optimal due to evolution and thus identifiable according to an appropriate criterion. We frame this as a metamathematical question, and using an order-theoretic criterion, prove that four qualitative conditions—monotonicity, convexity, continuity, and isomorphism—are sufficient to identify addition and multiplication over the real numbers uniquely from the uncountably infinite class of possible operations. Our results show that numbers and algebraic structure emerge from purely qualitative conditions, and as a construction of arithmetic, provide a rigorous explanation for why addition and multiplication are its fundamental operations. We argue that these conditions are preverbal psychological intuitions or principles of perceptual organization that are biologically based and shape how humans and nonhumans alike perceive the world. This is a Kantian view and suggests that arithmetic need not be regarded as an immutable truth of the universe but rather as a natural consequence of our perception. Algebraic structure may be inherent in the representations of the world formed by our perceptual system.

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Section: Convexitymentioning

confidence: 99%

The psychological scaffolding of arithmetic.

Grice¹,

Kemp²,

Morton³

et al. 2024

Psychological Review

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“…Natural addition is the minimum of all monotonic functions of the natural numbers. Thus, if humans are responsive to monotonicity (functions always increase), addition of whole numbers is, in a sense, the simplest operation to perform on them (Grice, Kemp, Morton, & Grace, 2020). Indeed, such additive systems are widely used in everyday life.…”

Section: Introductionmentioning

confidence: 99%

Range and distribution effects on number line placement

Kemp

Grice

Makarious

et al. 2021

Atten Percept Psychophys

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People's placement of numbers on number lines sometimes shows linear and sometimes compressive scaling. We investigated whether people's placement of numbers was affected by their range and distribution, as indicated by Parducci's (Psychological Review, 72, 407-418, 1965) range-frequency theory. Experiment 1 found large compressive effects when the endpoints were 1 and 10 16 . Experiment 2 showed compression when 14 logarithmically distributed numbers were placed on a line marked 1-1,000 and close to linear scaling when the numbers were linearly distributed. Thus, we found both range and frequency effects on compression. Where compression arose, it was not as pronounced as that predicted by logarithmic scaling, but analyses of the results from Experiments 1 and 2 indicate this was not explained by participants switching between linear and logarithmic scaling.

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The Psychological Scaffolding of Arithmetic

Cited by 2 publications

References 12 publications

The psychological scaffolding of arithmetic.

The psychological scaffolding of arithmetic.

Range and distribution effects on number line placement

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