The Cognitive Origins of Mathematics From the Origin of Counting to Infinite Series——An Operative-Genetic Interpretation of the Cognitive Source of "Infinitesimal"

Starting from the irreducible cognitive fact that “humans can count discrete objects and record their quantity,” this paper traces the cognitive origins of fundamental mathematical symbols—natural numbers, addition, subtraction, multiplication,division, fractions, square roots, and infinite series—through the chain of “operation→discovery of stability → recording.” The concept of “minimal unit transition” isintroduced to reveal how fractions and radical signs emerge from operational dilemmas in partitioning or arranging operations. It further argues that the summationof infinite series is not the addition of infinitely many terms, but a static naming ofan operation flow that “always leaves a remainder yet exhibits permanently stablepattern”; the so-called “infinitesimal” is in fact the operator’s record of the feelingthat “the endpoint is perpetually out of reach.” This paper does not attempt toreplace axiomatic systems, but offers an operation-based answer where axiomaticsystems choose to remain silent