[very] -- idempotence and reduplication
In mathematics, an operation * is called idempotent if x*x = x. For example, the integers 0 and 1 are idempotent under multiplication.
1*1 = 1
0*0 = 0
In linguistics, things are more complicated. George Boole claims that language is idempotent, i.e. that “red wine” is the same as “red red wine”. But is this true? After all, a “date” is very different than a “date date”. Repeated words, far from being unintelligible or logically redundant, actually serve as intensifiers. Idempotence does not exist in language. Instead, this special repetitive structure is called “reduplication”.
Music presents an interesting hybrid case. Melodically, music is idempotent. The note “a” sounded alone is the same as “a” played on top of itself.
<a a> = a
Things become more interesting when we consider rhythm. Quarter notes are held longer than eighth notes. Put another way, they have a greater “intensity” than eighth notes. In language, intensification occurs via reduplication. Duration can be thought of in exactly the same way. Longer durations are simply reduplications of shorter ones.
Although the natural numbers have been modeled using recursive structures in set theory, the more natural way to conceptualize them is through simple reduplication of the base number 1.