Metaphor is important in mathematics because it is the main way in which mathematical phenomena become more intelligible over time. They become embedded by successive metaphorical steps in a web of relations with other, better known phenomena. One fairly elementary example is given by the addition laws for the sine and cosine functions in trigonometry:
sin(a+b) = sin(a) cos(b) + cos(a) sin(b)These laws are not obvious and are difficult to remember. But when the exponential function is extended metaphorically to complex numbers by defining
cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
eiy = cos(y) + i sin(y)then the trigonometric addition laws follow immediately from the law of exponents:
ei(a+b) = eia eibby writing out both sides, carrying out the multiplication, and equating real and imaginary parts.
Here is a sample from the recent mathematics literature. Adrien Douady's 1966 thesis begins with the nesting of a rhetorical metaphor and a (exquisitely self-referential) mathematical one.
"Soit X un espace analytique complexe.|
Le but de ce travail est de munir son auteur du grade de docteur ès-sciences mathématiques et l'ensemble H(X) des sous-espaces analytiques de X d'une structure d'espace analytique."
( Let X be a complex analytic space. |
The goal of this work is to furnish its author with the degree of docteur ès-sciences mathématiques and the set H(X) of analytic subspaces of X with the structure of an analytic space. )