A rational analysis of the approximate number system

Area of Science:

• Cognitive Science
• Psychophysics
• Numerical Cognition

Background:

• Numerical cognition research shows higher numbers are less accurately represented than lower numbers, often modeled as a logarithmic mapping.
• Existing models derive this psychological law by assuming relative change in physical magnitude is key, but ideally, it should stem from more general principles.

Purpose of the Study:

• To derive the logarithmic number line from more general principles, specifically by minimizing representational error relative to the probability of needing each number.
• To establish a novel theoretical foundation for psychophysical scaling that relies on input probability.

Main Methods:

• Derived a logarithmic number line by minimizing the error between input and representation, weighted by the probability of needing each number.
• Quantified number 'need probability' using natural language data, aligning with probabilities found in other research areas.
• Developed a general framework showing optimal psychological scales vary with the square root of input probability.

Main Results:

• Demonstrated that a logarithmic number line minimizes representational error relative to number usage probability.
• Showed that this derivation does not require assumptions like Weber's law or specific internal representation forms.
• Proved that optimal psychological scales generally scale with the square root of input probability.

Conclusions:

• The logarithmic mapping of numbers is an optimal solution for minimizing representational error given the probability of encountering numbers.
• This approach provides a more fundamental derivation of psychophysical laws, applicable to power-law distributions and recovering logarithmic or power-law mappings.
• The findings suggest a universal principle for optimal psychological scaling based on input probability and representational efficiency.