**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
e^{iy} = cos(*y*) + i sin(*y*)

then the trigonometric addition laws follow immediately
from the law of exponents:
e^{i(a+b)} = e^{ia} e^{ib}

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. ) |

© copyright 1999, American Mathematical Society.