We don't learn numbers from set cardinality

A LessWrong article by user azergante challenges the common assumption that numbers are learned from set cardinality. Drawing on "Where Mathematics Comes From" (WMCF), the author notes that humans are born with an innate sense for small numbers and can subitize collections of up to 3 objects. Children learn larger integers by manipulating real-world collections (adding, removing, merging). While Zermelo-Fraenkel set theory (ZFC) defines numbers as sets of smaller numbers, a more intuitive mapping would be numbers as cardinality of sets. However, sets require distinct elements: a water molecule H-O-H has three atoms but the set of atoms is {H, O} with cardinality 2 — breaking the mapping. The correct mathematical object is a multiset, which allows duplicate elements and preserves the count of identical objects.

The author argues that our brains are pre-equipped with modules for tracking collections of objects (multisets), not abstract sets with uniqueness constraints. Therefore, building mathematics on multisets could be faster and more intuitive because it offloads reasoning to specialized neural wiring without the extra step of deduplication. While the author acknowledges practical considerations (e.g., whether foundations can be built on multisets), they cite the Wayne paper developing a multiset theory MST that contains classical set theory. The broader question: which mathematical objects let our brain run math programs fastest? The answer likely favors objects that mirror real-world collections, for which our brain has dedicated processing.