what is the scientific reason why peanut butter, mayonnaise, and
shortening removes gum from hair?

Why does oil float on water? Will sugar dissolve in gasoline?

The oil eats away at the gum.

Chewing gum has a polyisobutylene polymer backbone which is quite
hydrophobic (Greek for "afraid of water") but will stick to the
proteins in dry hair (note that gum does not stick to wet hair).  Oils
and fats are also hydrophobic substances, which "like" other
hydrophobic materials like the chewing gum. As the  oil from peanut
butter works its way into the interface between hair and gum, the
surface of the gum will prefer to stick to the oil rather than the
hair.
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