Multitude

Multitude is the quantity of a finite set. [TODO maybe circular because what do I mean by finite set?] Synonyms for multitude are natural number, whole quantity, cardinality, and the symbol $\mathbb {N}$; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.

Comparison

"Few" and "many" describe relative multitudes.

Counting

Counting is a process in which one identifies the multitude of a set, through iteratively identifying multitudes of its subsets.

Specific natural numbers

Multitude can be divided up into sub-concepts such as "1," "12," and "73."

1 is the multitude containing only a unit. [ostensive definition? is 1 a synonym for unit?]

2 is the multitude succeeding 1 in the counting algorithm.

3 is the multitude succeeding 2 in the counting algorithm.

4 is the multitude succeeding 3 in the counting algorithm.

Etc.

Addition

Addition is a process in which one identifies the multitude of a disjoint union, based his knowledge of the multitudes of its summands. (By the summands of a disjoint union $A\sqcup B$, I mean the set $A$ and the set $B$.)

Multiplication

Multiplication is a process in which one identifies the multitude of a cartesian product, based on his knowledge of the multitudes of its summands. (By the summands of a cartesian product $A\times B$, I mean the set $A$ and the set $B$.)