Origins of numbers: a shared language-of-thought for arithmetic and geometry?

Area of Science:

• Cognitive Science
• Number Cognition
• Mathematical Cognition

Background:

• Exact number concepts are traditionally linked to counting or approximate number systems (ANS).
• The precise cognitive and linguistic origins of exact numerical understanding remain debated.
• Existing theories do not fully account for the interplay between spatial and numerical reasoning.

Purpose of the Study:

• To propose a new theory for the origin of exact number concepts.
• To introduce a shared language-of-thought (LoT) model integrating geometry and arithmetic.
• To explain the emergence of integers and higher-level mathematical concepts through this integrated LoT.

Main Methods:

• Theoretical framework development proposing a shared LoT for geometry and arithmetic.
• Hypothesizing primitives of repetition, concatenation, and recursive embedding within this LoT.
• Utilizing the principle of Minimum Description Length (MDL) to explain mental manipulation ease.

Main Results:

• The proposed LoT engenders exact integers via recursive addition and multiplication from basic primitives.
• Connections between geometry and arithmetic within the LoT explain concepts like squares and primes.
• Mental representation efficiency (MDL) is hypothesized to correlate with ease of number manipulation.

Conclusions:

• Exact number concepts may originate from a shared geometric-arithmetic language-of-thought, not solely counting or ANS.
• This LoT model provides a unified explanation for basic integer formation and complex mathematical ideas.
• Preliminary evidence from historical, developmental, linguistic, and neuroimaging studies supports the proposed framework.