The psychological scaffolding of arithmetic

Area of Science:

• Cognitive Science
• Mathematical Biology
• Philosophy of Mathematics

Background:

• The origin and fundamental operations of arithmetic lack rigorous scientific explanation.
• Existing explanations from philosophy and cognitive science do not meet scientific standards.
• Biological origins of arithmetic, linked to adaptive behaviors like spatial navigation, are proposed.

Purpose of the Study:

• To provide a biologically grounded explanation for arithmetic's fundamental operations.
• To identify the evolutionary basis for addition and multiplication.
• To explore the emergence of numbers and algebraic structure from qualitative conditions.

Main Methods:

• Framed as a metamathematical question.
• Utilized an order-theoretic criterion.
• Proved sufficiency of four qualitative conditions (monotonicity, convexity, continuity, isomorphism) for identifying addition and multiplication.

Main Results:

• Addition and multiplication are uniquely identified from infinite possible operations using qualitative conditions.
• Numbers and algebraic structure emerge from these purely qualitative conditions.
• Demonstrated a rigorous explanation for why addition and multiplication are fundamental.

Conclusions:

• Arithmetic's fundamental operations are a natural consequence of biological perception, not immutable truths.
• Qualitative perceptual principles may be the biological basis for arithmetic.
• Algebraic structure might be inherent in the perceptual system's world representations.