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Metonymy and Metaphor in Mathematics

3. How to recognize a mathematical metaphor

In mathematics, metaphor occurs as translation of structure from one area to another. If the areas are close, you may hear ``similarly.'' Otherwise you will hear words like ``model,'' ``structure'' or even ``natural transformation'' when the translation mechanism is being studied as a part of mathematics.

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)

cos(a+b) = cos(a) cos(b) - sin(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

        eiy = cos(y) + i sin(y)  

then the trigonometric addition laws follow immediately from the law of exponents:

         ei(a+b) = eia eib  

by 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. )